The Point of the Banach-Tarski Theorem – not just a curiosity | Hacker News Reader

The Point of the Banach-Tarski Theorem – not just a curiosity

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76 pointsColinWright91 comments

Accepting or rejecting the axiom of choice as "natural" is a red herring based on the idea that mathematics express external truths about the world. Mathematics describe rules of reasoning which preserve truths about the world.

For instance, if I am trying to count my goats, the concept of natural integer allows me to count two groups, sum the results, and still have an accurate understanding of the number of goats I have.

As it turns out, the theory of natural integers allows one to define arbitrarily large numbers, even though I cannot possibly have more than, say, 1000 goats. Yet, a theory that deals only with numbers less than 1000 might be more cumbersome to work with. Try to define addition there and see for yourself.

The theory of natural numbers is useful because it provides a simple formal way to accurately count the number of goats (and possibly other things). It doesn't matter that it allows us to conceive of an absurd number of goats, like 1,000,000. It does the job, and as long you use it in the real world on real data you won't get absurd results.

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The same goes for the axiom of choice. Whether or not you accept it isn't going to make a practical difference once it comes to measuring the volume of helium needed to fill your balloon.

Think of it as upcasting and downcasting. You're dealing with a finite problem which is hard to solve. So you upcast it to a more general theory, solve it, and then downcast it back to the original setting.

There are as many ways to upcast as there are ways to pick independent axioms, so pick whichever make the problem easiest to solve, and don't worry about the "truth" of these propositions.

This "upcasting" and "downcasting" also work with mathematics itself. For example, we can consider adding infinitesimals to the real numbers, allowing us to do calculus without taking limits. Then we can downcast back to the real numbers.

The problem is that some concepts might not have an "isomorphism" to the real world (as Hofstadter calls it). In fact, finitists believe that the concept of infinity possibly introduces some discrepancy, making the "casting" a lossy transformation.

Exactly,"upcasts" can be one-way

Do you have an example?

Presumably the axiom of choice is a good example of something unlikely to be applicable in a real world context. Unless you're trying to anticipate when a rogue mathematician is about to attack, of course.

(Also halting oracles.)

Another perspective is that mathematics is not intended to describe the real world, and so what makes a mathematical assumption "natural" is how elegant or inelegant its consequences are. There is a long list of strange things that would be true if you assume the axiom of choice is false [1], and I'm pretty sure nobody who has ever considered the pros and cons of AoC has cared about the amount of helium in their balloon. Esp. considering that it's not possible to construct nonmeasurable sets in the physical world (since you would need to do operations at a scale infinitely smaller than the Planck constant, for one!).

[1]: http://mathoverflow.net/questions/129036/counterintuitive-co...

The reason those things are strange is that uncountable sets have absolutely no bearing on computable objects (== enduring in math and science) but everyone mistakenly applies their countable intuition to uncountable set theory.

If one uses classical logic there are even many countable, even finite sets which are beyond computable distinction. The set of computable functions on say natural numbers is a countable set, but even distinguishing when two such are equal cannot be effectively done.

The solution is either to distinguish between countable and enumerable and decidable, or to use intuitionistic logic.

> Another perspective is that mathematics is not intended to describe the real world

Isn't that a fairly recent perspective, though? My memory is that the concept of math for its own sake (purposefully divorced from "real world" applications) is only a couple hundred years old at least. Whereas for most of its history, from the Egyptians up to Newton, mathematics was developed as a tool to understand the physical world.

So while some people today might consider that mathematics is not intended to describe the real world, they're working on top of a deep system that was.

> what makes a mathematical assumption "natural" is how elegant or inelegant its consequences are

So now we start to argue about concepts that are not well defined in the context, introducing new terms, even?

Art for art's sake is just one aspect of science and craft. You might argue that mathematical knowledge as it is incorporated in the material world, e.g. in brains, books and soundwaves, has become a real matter which entails all the complexity that makes it so difficult to grasp.

> introducing new terms, even?

We do, but you have to remember that mathematics is an activity invented by humans largely for their own amusement. This is especially true when it gets down to the foundations of mathematics.

So let me put it this way. Your description of mathematics is not very amusing to me.

>Mathematics describe rules of reasoning which preserve truths about the world.

I'd disagree even with this, and state something even weaker. Mathematics describes rules of reasoning. A sound mathemathical theory of the something in the real world is a (partial) correspondence between statements about the real world and mathematical statements, such that the mathematical manipulation of statements preserves correspondence to truths about the world, or produces statements that don't correspond to the world.

I think this distinction is important because it emphasizes that it is the scientists' responsibility to pick an appropriate mathemathical model.

Zermalo-Frenkel Set theory with the Axiom of Choice is not proven to be consistent. One might argue that consistency is a natural requirement for a rule to be a rule. Especially for peano axioms, consistency is a given, so your comparison doesn't hold. Following the incomplefeness-theorem, the (dis-)proof might be external to any of ZFC's descriptions of the world, which believing in ZFC would be the world.

disclaimer: I don't understand axiom of choice.

The author of Irregular Webcomic walks through a Banach-Tarski decomposition and it turns out not to require any really heavy-duty math. You should understand it if you know what rational and irrational numbers are:

http://www.irregularwebcomic.net/2339.html

"I have discovered a truly remarkable proof of the Banach-Tarski theorem which this margin is too small to contain - oh wait, this is a webcomic, of course there's room!"

I just want to mention that there is a much deeper discussion to be had about the axiom of choice than the apparent impossibility of Banach Tarski, and it has to do with equality.

In mathematics is is common to change the equality between sets (form quotient sets). For example, the set of real numbers are infinite sequences of rational numbers (fractions) which approximate the value. But the equality which we place on this set is not the equality of sequences, which is too strict. We put a new equality on these real numbers which identify two of them iff they come arbitrarily close to each other.

The axiom of choice (AC) says that if for a ∈ A there is b ∈ B satisfying some requirement, then there is a function A → B selecting these elements. The axiom of choice is only problematic if we require that choice function it produces respects some quotient equality we placed on A and B. The confusion arise because ZF Set Theory cannot express the weaker notion of a function which does not express the quotient equality. Other foundations, such as Martin-Löf Type Theory, can express this difference and you get AC without the paradoxes.

What's the disadvantage of those other foundations that makes them unsuitable for universal use?

Of course the proponents of these other foundations claim that they are suitable for universal use. But sets came first, and is good enough for most purposes, and mathematicians in general are more concerned with mathematics than foundations. Actually the recent movement of "Homotopy Type Theory" is gaining some traction among non-logicians, giving easier formulations and proofs in category theory and topology.

Also in computer science type theory is becoming more usual, since all results are constructive (provides algorithms) and is in fact a programming language in it self.

They're certainly not unsuitable, but they have a tendency to take observation/realization/grasping/proving as an instantaneous process. In most people's regular understanding of logic, however, this is false. There is a process by which new data and new conclusions from that data are drawn and made available to you.

So ultimately, if those things matter for either UX of mathematics or for actual mathematical content is a large question, but it's fairly clear that if you need to capture something closer to the actual dynamics of knowledge/learning/understanding then you are sunk when you base your foundations in set theory.

I don't think there is any confusion here. Of course ZF set theory can express this weaker notion. But in ZF these different functions have different domains and ranges A, A mod r, B, B mod r, where r is the equivalence relation.

In my experience it is type theory which introduces A LOT of confusion. Like the above confused comment.

I am not saying ZF cannot distinguish functions which respect an equivalence relation and those which do not — the axiom of choice concerns equality, not any old equivalence relation.

I am saying that ZF cannot distinguish those sets for which the logical "for all x there exists a y" defines a function. This is because logical equality in ZF can be forced to be be any equivalence relation, without any additional restriction on the logical constructions. This forces us to add extra axioms saying which sets are ok: countable choice, dependent choice etc. At the extreme end there is the axiom saying all sets have this property, with consequences such as Banach-Tarski and Diaconescu's theorem.

Type theory introduces nuances, which may not be appreciated by the non-logician (constructiveness, intensionality…). Homotopy type theory goes some way in mitigating this by allowing quotients (but the logical constructions are suitably restricted to preserve choice), and adding function extensionality (not to mention univalence).

And ZF should'nt distinguish those sets for which the logical "for all x there exists a y" defines a function or not. If you want the notion of a unique function, you should have the logical statement "for all x there exists a unique x".

I have no problem with Banach-Tarski. Sets of points just do not describe properly our intuition of 3-dimensional space. Measurable sets of points more so.

Type theory just shuts out certain things its inventors don't want to think about. How does that help anyone?

The intuition behind the type theoretical axiom of choice is that the choice is given by a (hypothetical) proof of the "∀x ∈ A ∃ y …"-statement. The constructive interpretation of the statement is precisely that there is such a choice function — so it becomes a tautology (provable without any extra axioms). And I believe that agrees with the naive intuition one has why the axiom of choice is true. The problem is when the logic allows proofs (or semantics) which does not preserve the equality. In type theory the equality is intensional and thus preserved by structure.

Unique choice is besides the point here. There may be a unique function, or there may be many. The intuition of choosing is the same.

I also have no issue with B-T, as I expect weird consequences of ZFC; the axioms and logic being so profoundly non-constructive and removed from my intuitions in the first place. That said, I would not deny the set-theorists enjoying their theory, I merely observe that there are subtle things their theories fail to describe.

Type theory does not shout anything, but is carefully constructed from the needs of its creators. Some people want to extract algorithms from their proofs, or want to prove that their algorithms are correct. And if you want this, the nuances of type theory are actually nuances of things you care about — algorithms and proofs. If these are not among your interests, then type theory is not for you. Thats fine, I am sure you have other interests.

There is no evidence that the real world has anything to do with a continuum, real numbers, or R^3. When we look closely at things, we always find combinatorial structures of discrete entities (smooth matter -> atoms -> protons -> quarks). For example, electrical charge is discrete, not continuous, and the unit appears to be 1/3 the charge on the electron (as carried by quarks).

The balance of evidence is that our universe is spatially compact, with time bounded in the past (Big Bang), so it is possible that there are no infinities at the cosmological scale either. The evidence for the temporal future has recently changed from closed (Big Crunch) to open (infinite expansion).

There are various theories about how space and time emerge from discrete quantum mechanical structures. For example, a simple derivation in Loop Quantum Gravity gives a discrete spectrum for area. At the moment, we don't know which theory is correct, but it is likely there is a smallest finite unit of area (and volume).

Another reason for rejecting real numbers in physics, is that each value contains an infinite amount of information. The Holographic Principle is interesting because it rejects the notion of an infinite amount of information in a finite space. In fact, it goes a lot further, and says the total information within a 3D volume does not scale as the volume (!), but as the 2D surface area. Essentially there is one bit per Planck Area, so 10^66 bits/m^2.

It is easy to imagine, but not to describe or prove, a combination of these theories that implies everything is finite, discrete and combinatorial, without an infinity or a real number in sight.

But that's not really the point.

    All models are wrong,
    some models are useful.

Using the reals as a model makes the math a lot easier than trying to do everything in discrete structures, especially since we don't actually know the scales that these discrete structures might be at. We don't know what the smallest unit of space is, and it's still not clear that there is one. As a result we use the reals as a model, and it does incredibly well.

So in a sense you are helping support the main point of the original article. The fact that the reals are an approximation to what you claim reality might be should be acknowledged, and then we should explore the limits and limitations of that models. The Banach-Tarski theorem does exactly that, showing us at least one place there the model apparently fails, and should certainly be treated with care.

>So if we have a bounded set (and I've not given a technical definition of what that means) then we'd like the measure of that to exist

Why in the world should this be true? I'd much sooner try to weaken this requirement than the additivity requirement or the axiom of choice. For example, I imagine restricting measures to (unions of countably many disjoint) (path-)connected subspaces of the ambient topology would solve the problem just as handily, without making any compromises on how well the math models our intuitive notion of measure. On that note, I suspect that to really have a meaningful notion of measure, one really needs to have at least a topology, if not an outright metric space.

Can anyone more versed in this area comment on this?

People I talk to about this are often very unhappy about countable additivity, and are often unhappy about infinite in general. I suspect that the reason they think they want a measure to be defined on every set is because they don't know just how weird sets can be, and are relying on their intuition. Everything can be weighed, everything can be dunked in water to test its volume, so why should something not have a "volume"?

You clearly have a different intuition born of your background and training. You're talking about

  > "... unions of countably many disjoint
  > (path-)connected subspaces of the ambient
  > topology ..."

You already have an unusual intuition.

And you're right - it's "obviously" better to restrict the sets we play with rather than limiting ourselves to finite collections, but that "obvious" comes from years, perhaps decades, of playing with these ideas.

And with regards having a topology or a metric, consider that perhaps a measure can be coerced into providing a metric ...

> Everything can be weighed, everything can be dunked in water to test its volume, so why should something not have a "volume"?

But it seems pretty obvious to me that you can't measure the volume of something (accurately) if it has features smaller than a water molecule.

Don't know about an equivalent example for weighing things.

But it stands to reason that, whatever you (think you) want to measure shouldn't have meaningful features smaller than the accuracy of the thing you're measuring with.

That's why we made electron microscopes, because photons were too big.

A lot of the applications of measure theory are in fourier analysis and probability, where countable sums are necessary, and topology often isn't. Also, just because the topology on R yields a certain intuition doesn't mean it's the right way to generalize the theory.

> If that's true, then there can be no measure satisfying the requirements even when weakened from countable additivity to finite additivity.

I think this is an important/interesting point and should be highlighted more in the article. The "magic trick" is not hidden in the fact that you are using infinitely many pieces. The "magic trick" is that the pieces have very odd shapes.

OK, I've made a change to highlight that, and it's processing and uploading now. Let me know what you think.

Specifically, shapes with infinitely many "sides" in a bounded region. (a ball)

I am very much not a mathematician. I took physics at university, but that was 20 years ago. The mathematics here is beyond me.

So, I realise I'm probably confusing theoretical mathematics with a practical science. But, still. If this was a real, true result, couldn't someone simply take a sphere of, say, gold, and recut it to be two spheres of gold? Then keep doing it until they're richer than Croesus?

Or, to be more serious, what value are results like this if they are obviously false in the real world?

Also, if they plainly don't apply to the real world, isn't that a really good sign that the mathematics is actually incorrect?

Or, have I completely misunderstood the article?

You would need to cut the spheres in very strange pieces and since spheres of gold are made of atoms you can't do that in the real world.

Why not, assuming you make the spheres big enough? Let's say not gold, but a moon. I mean, let's do a thought experiment with superhuman tech and a big enough sphere that atoms are small enough to make the slices right.

I guess I just don't understand why this doesn't violate conservation of mass.

Because you really, really can't make the slices small enough.

And your point about violating the conservation of mass is the whole point of the article. When we try to come up with a mathematical model of what "volume" means there are certain properties we want it to have. One of them is that any arbitrary collection of point - not atoms, but mathematical points - can have a volume associated with it. The Banach-Tarski theorem shows that such a requirement is impossible.

You're not alone if you think this is all nonsense - so did Feynman. However, many clever people not only believe that this is relevant, but also useful and insightful. The article is trying to give a sense of why that's the case.

I want to write a sister article to this to help people like you come to understand what's going on, but I'm having trouble finding people who are willing to engage with me on it. They usually just find that it offends their sense of reality and reject it all. I think that's a shame, because unless people like me can come to understand what others find so objectionable we can never learn how to help people understand why this is interesting, useful, adn relevant.

I think the problem is there's mathematics on one hand and my experience of the real world on the other. I either didn't do any set theory, or I've forgotten it. So it looks just like squiggles on the page to me when I try to understand the maths.

So, all I'm left with is trying to relate what the words might mean in real-world terms. And coming up confused.

I don't think it's your fault. I just lack the grounding to see both sides of the picture.

One thought, though. I'm not trying to reject it because it offends my sense of reality. I'm trying to use my sense of reality (which is the only tool I have) to understand it.

For example, if someone came up to me at work and said, "I've just worked out how to cut a sphere up into bits and reassemble it as two spheres the same size," I'd say, "ok, then, show me."

If all they could do is make marks on paper I couldn't understand I'd think they'd got the paper wrong, not reality. Which is the dissonance here. I guess, speaking personally, if you want someone like me to understand it you'd need to really, really explain the maths (as if to a simpleton!), or explain what's happening in real-world terms and why it wouldn't work yet is still valid.

Does that help?

This book gives a good explanation of a construction, at a reasonably non-technical level.

http://www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/15...

But what we're saying here is this.

We use sets to model the real world. They do remarkably well, and the math we've developed to work on them includes calculus to make bridges that stay up, fluid dynamics that make aeroplanes fly, and discrete math that helps us understand routing, scheduling, and all sorts of stuff.

We use the real line to model distances. In the real world we are limited as to the accuracy we can use, but modelling those limitations is nasty. It's easier to assume that things are continuous. In its turn, we make choices that make working with these models easier, and they turn out to be amazingly useful.

But then we start poking the dusty corners. The choices we make in the development of the theory have consequences, and math is about exploring both the choices, and their consequences.

So we can choose that between every two numbers there's another number. We can choose that there's a number whose square is 2. We can choose that the sum of the reciprocals of squares : 1+1/4+1/9+1/16+1/25+1/36+ ... : is a real number.

And we can choose that there is no smallest positive number. That has a consequence. If you believe that there is no smallest positive number, then 0.99999... has to equal 1. You can't have one without the other.

So we can talk about "the length of a line." Then we can talk about the "length" of a set of points on a line. Then we start to find that these simplistic models, these obvious and natural choices, even though they are amazingly useful have some unexpected consequences.

Does that help you to understand the context?

I'd be really interested in developing an agreed dialog about this. Will you send me an email?

> I'd be really interested in developing an agreed dialog about this. Will you send me an email?

I've sent you an email, as requested.

From the sound of it, though, the answer to my puzzlement, is that it doesn't apply to this universe. In which case, I don't think it's a problem at all. I'm quite happy to imagine mathematicians doing work on geometries that don't map to the real world, for example.

Mostly, then, is this just a question of presentation? I mean, if you said "this does not apply to the real world, but to some fictitious mathematical assumption that assumes things can be divided up infinitely," I don't think anyone would have an issue with it. It sounds like it's a problem to most people because it's presented as if it's a real-world result.

Or, at least, that seems to be the impression I get, given the other comments.

What I mean is, is there's a question of misdirection here? I.e. the theorem is presented as this non-intuitive thing that can't possibly be the case in the real world, then when someone asks what would happen if you tried it in the real world, the answer is "it's assuming some things that aren't true for the real world." Because if that's the case, I'm not sure I see the paradox. Assuming a weird set of ideas, I would expect you can come up with weird answers.

Here's a question, though, because something is nagging at me. And I'm going to assume this is a universe of infinite points and no atoms (as I understand it, at least).

Let's say we have a sphere of volume 4/3 pi r^3 = 100

Now you do your cuts, but don't reassemble yet. The sphere is still the original sphere, with all the shapes it has been cut into still in virtually their original spaces.

The total volume still has to be 100, right? I mean they all still fit into the original space.

So now, you immerse it in water, in a bathtub ready to overflow, and start manipulating the pieces.

At what point does the water level rise?

I've replied to your email, but to reply here to your specific question:

  > immerse it in water, in a bathtub ready
  > to overflow, and start manipulating the
  > pieces.

  > At what point does the water level rise?

A lovely question. The answer is that the "water" isn't "water", it has to be the same infinitely fine "mush" that the ball is made of. As a result, as you move the pieces out so the "water" ends up forming non-measurable holes to fit them into, and the complement is also non-measurable. So in the same way as the balls kind of "fold out" to become two balls, so the water kind of "folds in" and takes up less space than it used to, exactly balancing the actions of the pieces.

The four pieces don't have measure, don't have unambiguous volume. It might help to think of them as fractals (though in fact they're even more ambiguous than that).

The only "realistic" answer I can give is that each of the four pieces would "take up" the full volume of the sphere as you moved them apart. Each of the four pieces is "shaped" like the whole sphere, but with infinitesimally small holes where the other four go. So not a single "water droplet" would ever go inside the four pieces, and while you were moving them around they'd take up the space of four of the original sphere.

How are they more ambiguous than fractals? The key structure is a shape that can be rotated onto a subset of itself.

I guess with common fractals, you camnoly scale an object onto a subset of itself. The magic of BT is that the ball-shape allows a rotation that looks like a scaling.

A fractal tends to be kind of contiguous - for most points in the fractal there's a neighbourhood of those points that's also in the fractal, and similarly for points not in the fractal. These are "dust sets", dense everywhere in the sphere and their complements also dense.

Fractals generally have a formula describing them, i.e. they are computable. Nonmeasurable sets are necessarily non-computable. This is why you will never see an accurate picture of the B-T sets, because they are not defined uniquely by any formula or algorithm (which is connected to the fact that the axiom of choice is needed to even prove these sets exist).

Your final question is very interesting, it gets at the heart of the matter: the Banach-Tarski theorem shows us that in your universe of infinite points and no atoms, there does not exist a definition of volume which is consistent. So in this universe the water level in a tub cannot be defined.

You at exactly right. The "paradox" is that people naively assume that the Real ("Real" = math theory, "real" = CS turing-computable and physics "real universe") numbers are a smooth continuum, but if you follow the actual definition, you discover that it is too powerful -- the Real numbers could construct physically impossible objects, which proves that real numbera aren't real. In reality, we must confine ourselves to countable sets, which has ugly asymmetries: we must distinguish a set of "nameable" numbers as more "real" than the others, but we can choose any small-enough subset of Real numbers we want, we can choose every single Real number we practically encounter, but we can't choose all of them at the same time.

I'm not sure in what sense you mean real numbers could "construct" anything, but you can make physically impossible shapes out of rational numbers too, so their cardinality has nothing to do with that.

One way to look at this is to note ask what exactly it meant when you said "a sphere of volume". In the real world it's ultimately impossible to imagine spaces which don't have volume of some kind or another, but in mathematics we often study models of the world too sparse to even have "measurability". These are obviously aphysical, but it turns out that when one tries to construct reality via set theory you have to go through a zone of "funky aphysical things" before you reject them all and consider only measurable things.

BT exists just before we make that last transition and abuses the fact that the constructions we've made so far are not required to have any reasonable sense of "volume" whatsoever. All of the "infinity" bits are sort of a red herring as there's no reasonable way to think about taking physical things and cutting them into aphysical things. Instead, this all arises from taking incredibly aphysical things (raw, theoretical sets) and building them up until they feel physical.

If you lay out the constraint that "cuts" must divide measurable things into measurable things, which is a reasonable expectation we have of physical items, then BT will vanish.

Actually, reading the comic another poster put up here:

http://www.irregularwebcomic.net/2339.html

... I think I've managed to get my head round it. Or, at least form an initial model.

It's like the old puzzle of the hotel with infinite rooms, each one labelled with an integer.

Then, another infinite set of guests turn up. The hotel manager asks every person currently in the hotel to double their room number and go to that one. Because there are infinite even numbers, this works fine.

Then, the new set of infinite guests sleep in the odd number rooms.

If I'm right, of course, then this only works because infinity is really odd - and you couldn't get an infinity room hotel in the real world.

Is this a reasonable way of thinking about it?

Short answer, yes that's a good way to think about it. The Hilbert Hotel paradox is closely related to the B-T paradox, it only adds a few more complications to make a stronger statement.

The key step in the B-T paradox is setting up a two-dimenional free group in just three-dimensional rotations. This is an infinite discrete set, a sort of analogue of the integers (which is the one-dimensional free group). In the integers it is no surprise (or maybe it is) that you can take the positive integers and shift them all to the right by one unit and then suddenly you have made a "hole" from apparently nothing (the Hilbert's hotel paradox). One worrisome part of this is that the "rooms" must get infinitely far away, but B-T exploits the fact that by using irrational rotations you an wrap the whole construction into the space taken up by a single sphere.

Add just a little more mathematical mumbo jumbo and you have a two-dimensional free group, which is a hotel such that every room is as big as all the rest of the rooms and you can set up the hotel so that each room is as big as all the rest of the rooms, and then you can do magic such that a move to the left actually leaves as much space as the whole hotel. And since it's all wrapped up in a ball it doesn't even run off to infinity to do this.

You do need the rooms to be points with zero volume, though; otherwise even with one-atom rooms you still need to shove the entire infinite hotel into a finite space and you will run out of room.

You could make the same argument against the real numbers. Almost all of them cannot even be written down or described, so it's hard to say how they map to the real-world. But the concept of real numbers is extremely useful for a large number of mathematical proofs.

Which proofs depend on an infinitely discontinuous subset, though?

It's not that we rely on such things, but that these things are unavoidable consequences of choices we make that seem to be perfectly reasonable, and result in useful math.

Phrasing in the other way:

As we develop math that we find useful and powerful, we find that we have to make choices. Those choices have consequences, and sometimes as we explore the consequences we find that really strange things happen.

We can go back and make different choices, but in practice we tend to find that no matter what choices we make there are odd and hairy things that result.

The natural way to encode an infinite binary tree into the real numbers is as the Cantor set, an uncountable totally disconnected set.

Uncountable sets are not reality. They are unicorns. They are a pretend construction that cannot be mapped 1:1 to anything in the universe, by definition. "Uncountable" means "outside the realm of real world algorithms and physics" . they are like "god of the gaps", a name for what we call the beyond the edge of the universe, which we can never reach. But we can assign some structure to it, roughly as an extension of reality, to imagine what it could be like. Sort of like heaven. Sort of.

As far as we know, you cannot cut a fractal such as the Mandelbrot set out of anything physical, either. The details get infinitely fine, and that's not possible in physical objects (as far as we know)

Conservation of mass is a consequence of the fact that matter is not infinitely divisible. The math on this works out too: there is an easy "measure" on three dimensional objects in the real world - just count the atoms. (This is known as a "counting measure" in measure theory.) It is the presence of infinities in the real numbers that causes this measure to fail (you will end up measuring the B-T sets, and regular objects like spheres as well, to have infinite "mass", so that 2*infinity = infinity is no longer as surprising a result).

Why would the strangeness matter? The article insists that it's not an issue of infinitely small pieces, so what's the actual inherent barrier to doing this on real objects?

Is it an issue of continuity that's not related to the pieces being arbitrarily small? I still don't see where this refutes the traditional line about the theorem being an artifact of uncountable sets.

It's not that the cuts are strange, it's that they involve infinitely small details. You can't cut physical atoms like that, and so the construction fails in the real world.

Then his article just leaves me more confused than before. I was already familiar with the traditional resolution of BT as "well, that's just an artifact if the weirdness when you have uncountably many points".

But now the author insists that "oh no, you get the same paradox with finite pieces". And yet on every probe of that point, it comes back to an issue of infinities. So what's wrong with the traditional explanation? And how does this article justify a "finite version" of the partition.

I don't understand what you think the "traditional resolution" might be. In the Banach-Tarski theorem you are partitioning a 3-dimensional solid ball into finitely many pieces. Because the ball has uncountably many points, those pieces will have uncountably many points.[0]

Does that help?

[0] Actually that only shows that at least one of the pieces must have uncountably many points, but in the theorem we find that at least four pieces must have uncountably many points.

The article doesn't try to explain how the construction works. The construction is done with finitely many pieces (I think five pieces is the limit), but those pieces have infinitesimally small details.

Having finitely many pieces is different from finitely many operations you would need to cut the pieces in real life.

For example, there is a curve that cuts the plane into just two pieces, but the line itself has infinitely small resolution [1] and hence you can't cut it. In fact, there is no segment of this line, no matter how small, that you can cut in a finite amount of time!

[1]: http://en.wikipedia.org/wiki/Weierstrass_function

The subsets you partition in are necessarily non-measurable sets. That makes them very strange. Any reasonble set that you can come up with is measurable. Even the set of points with rational coordinates is measurable. Of course you cannot partition a gold cube into two pieces, one with the points that have rational coordinates, and the other with points that have irrational coordinates. Non-measurable sets are even weirder.

This is only an indication that arbitrary subsets of R^3 are not a good way to model three dimensional space.

It is a real, true, result, in a space that is based on different rules than our physical reality. A ball in R^3 is composed of a countably infinite number of zero-dimensional points. It's an abstract concept that doesn't really have much to do with a physical ball that consists of a finite number of atoms.

Fundamentally, math is something that describes a large collection of possible worlds. Our universe is just a small subset of all the things that could be, and when using math to understand reality we have to remember that.

I'm pretty sure you mean "uncountably infinite" instead of "countably infinite".

Yep, thanks!

Not "possible", " imaginable". "Possible" is begging the question.

Have you ever heard stories like "Hilbert's Hotel"?

Given an infinite hotel (each room labelled with a positive integer), you can empty half your rooms by moving the person in room x to the room 2*x, which gives you all the odd rooms to fill with a new infinity of people, effectively doubling the size of your infinite hotel?

This is very similar -- except it works over the real numbers in 3D space. In particular the cuts are "inifinitely fine". Given any point p, and any distance d, there will be a point less than distance d from p which is in a different "slice".

Imagine (while this isn't in banack-tarski) making a "slice" which is all numbers of the form 1/x for all integers. Clearly this "slice" could really exist, but mathematics can of course define it and operate on it.

Rudy Rucker explored this thoroughly in his novel "White Light" (which is where I first learned of the Banach-Tarski theorem) whilst having with much fun with infinities.

It's finitely many pieces, but not in the way you think. Each supposed piece isn't something you could carve with a knife, even if we defined "carving" as a mathematical operation. It's just a set of points with coordinates that obey a certain rule.

In mathematics you can say things like "the set of all points in the sphere with irrational coordinates" and think of that as one "piece" of the sphere but it doesn't correspond to anything in the real world.

I'm not a mathematician either, but I think the consequence is just that mathematical points are rather weird and admitting any infinite set of points (however disconnected) as a volume produces weird results. What we think of as a "piece" isn't just an infinite set of points.

The result IS true, just like "2i*2i=-4" is true. It just doesn't apply to physical reality, at least not in the most straightforward way (just like you can't have 2i chests with 2i apples in each chest).

What if you have 2 iPhones?

Banach-Tarski is absurd because matter in reality is made out of particles. If you take this into account in your definitions you can't do absurd stuff like Banach-Tarski.

As such it offers an insight into what can happen if your mathematical model fails at representing the reality.

Strictly speaking, math it self isn't intended to reflect reality. It's just logical system together with some initial axioms. But it does happen to model reality very well, given the right interpretations.

But uncountable sets do NOT model reality well.

Oh but they do. When taking measurements in a scientific setting, it usually gives the most accurate and simple model to assume that your measurement is an approximation of a certain real number, and understand more accurate measurements as better estimates of this real number. Using rational numbers or other countable sets in this position usually leads to undesirable biases and or circuity in the model. When you need a "continuum", the real numbers have the best properties for the job.

Let's say you've got a light-emitting flat surface and want to describe its power output. I recommend modeling it as a set of real coordinates with an intensity value at each point and integrating to compute the total power output. By all means I invite you to come up with a better model, without using uncountable sets, for which there isn't a straightforward analogue, just as good, that uses uncountable sets.

Edit: Might as well jump straight to explaining how wave functions work without using uncountable sets, and why that's a better description of reality.

Banach-Tarski is not absurd. Supposing the world should behave in a certain way because there is a mathematical theorem that seems to say it should, that is what is absurd.

Banach-Tarski is important and interesting, because it neatly illustrates why mathematics that seems to be sensibly related to reality sometimes does not describe reality very well. Relations between mathematics and reality are tricky.

BTW, the finiteness of particles is not sufficient to explain the problem. E.g. electrons and photons do not have a finite size. Yet you cannot something similar to a ball of photons or electrons either.

It's not just that the particles have to be zero size. There also have to be infinitely many. Otherwise, you could just count before and after and apply conservation of mass. (This also applies in mathematics: any finite set has its size preserved by euclidean motions, unless you put points on top of each other.) Needless to say, gathering an infinite number of electrons or protons will cause black hole-themed "reality breakdown" problems long before you get there.

Excellent article, but my first reaction was "Goodness, that's an awfully punny title". It still gives me fuzzies, but I can't tell if it's intentional or not.

If one solid ball can be partitioned into two solid balls of equal radius, then can't each of THOSE be partitioned into four solid balls of equal radius, and so on ad infinitum?

Yes. Except that the ball can't really "be paritioned". That's a mathematical illusion. There is an isomorphism of structures, but there is no computable algorithm that can perform the partition operation.

Maybe it helps to circularize the problem a bit? Or at least, if not help, not make it worse? And then mix in some quadratic equations?

http://en.wikipedia.org/wiki/Non-measurable_set#Consistent_d...

If B-T is true, then you've got to select 1 of 4 known cheat codes to allow the definition of volume of "normal real world things" at least the way non-math people like to measure volumes. The least icky is the option that demands sets with non-measurable volumes exist, weird as that sounds. Then again, how weird is it really, given that no one freaks out about an infinite number of irrational numbers existing in between all possible fractions. Pi, after all, is no fraction, but its handy to keep around anyway. If you assume B-T is true and in choose-your-own-adventure fashion select that non-measurable sets exist, then, it turns out that having non-measurable sets make B-T "obvious-ish" or at least less obscure sounding, it all kind of works out in a circular manner.

Why in the name of Occams Razor would you want a pair of weird ideas instead of dust bin both and stick to grade school geometry? Well the axiom of choice wedges in sideways between B-T and the non-measurable sets above, kind of like two balls wedge into the space one ball takes up above (making kind of a joke or tongue in cheek). Its not just two weirdo ideas that work together but a couple of them. And the axiom of choice is just so useful in so many ways (see its wikipedia page) I'd have to think for a second about chucking out the axiom of choice. I think it would be ickier than keeping it around. Life is so much easier if you keep all three hanging around, and all their hangers on.

A really good analogy would be some real world quadratic equations for a land survey (well, made up example) only have one solution in the reals although everyone knows there's two mathematical solutions to any quadratic even if you don't like negative sq roots, and thats OK, and you kinda have to look sideways at the solutions involving negative square roots. Its not that real world geometry problems are full of negative square roots in practice or it means anything in the context of land survey problems, but its kind of a place holder in math.

Kind of a "conservation of weirdness" physics theory where they cancel out over a large enough collection of theorems or a collection of weird ideas is in sum less weird than any individual idea. So if you'd like this and that, and its really interesting and handy and seems to work quite well, sometimes you're going to have to not look overly closely at weird point sets that literally do not have a defined volume, at least not as you'd define volumes, and then screwing around with those volumeless objects can result in super weird stuff like two balls for the price of one. Which is OK in the physical world because we don't have abstract spheres that can have anything happen to them, we have vaguely round piles of atoms with really complicated rules about what you can do to that pile of atoms.

It all vaguely resembles the manufacture of sausage where you'll probably be happier if you don't look to closely at things that shouldn't exist, yet, its a tasty breakfast sausage if you don't think too hard about where any individual part came from. This is a highly heretical view, we're only supposed to think about math as some beautiful, pure, and virtuous thing, which I'm convinced is sociologically some repressed Victorian views about virgin brides or some nonsense so I don't feel too bad about being a heretic.

> Pi, after all, is no fraction, but its handy to keep around anyway.

You actually only need a surprisingly small amount of decimals of pi to calculate the circumference of the visible universe (just about the largest circle you can possibly have) to the accuracy of a single proton (just about the smallest scale you can realistically want to measure).

I think it was about 50 decimals or so.

Take that, crazy hundreds-of-decimals-of-pi memorizing people! ;-)

I don't believe in the axiom of choice.

You don't believe it is true? That's silly, there is no good reason to believe it true or false, except in the sense that you think one of Chess or Go is the more fun game of the two.

Or you don't believe it is a meaningful to try to choose an answer? That's ZF, the home to the portion of set theory that has anything useful to say about the physical Universe.

So because Go is (objectively) more fun than Chess, that means Banach-Tarski is bollocks!

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