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

http://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?HN_20150607

by ColinWright • 11 years ago

76 91 11 years ago

solipsys.co.uk

gus_massa• 11 years ago

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.

ColinWright8• 11 years ago

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

reagency10• 11 years ago

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

summerdown25• 11 years ago

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?

gdsimoes• 11 years ago

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.

summerdown25• 11 years ago

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.

SilasX2• 11 years ago

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.

Sharlin2• 11 years ago

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.

dietrichepp2• 11 years ago

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

reagency10• 11 years ago

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

CJefferson• 11 years ago

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.

mikro2nd• 11 years ago

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.

skybrian• 11 years ago

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.

ajuc• 11 years ago

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).

tripzilch4• 11 years ago

What if you have 2 iPhones?

legulere• 11 years ago

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.

backprojection• 11 years ago

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.

reagency10• 11 years ago

But uncountable sets do NOT model reality well.

Confusion• 11 years ago

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.

digama5• 11 years ago

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.

murbard22• 11 years ago

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.

---

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.

dietrichepp2• 11 years ago

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.

rndn• 11 years ago

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.

platz• 11 years ago

Exactly,"upcasts" can be one-way

murbard22• 11 years ago

Do you have an example?

j2kun3• 11 years ago

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...

reagency10• 11 years ago

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.

snowwrestler• 11 years ago

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.

fusername• 11 years ago

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.

im3w1l• 11 years ago

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.

musername2• 11 years ago

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.

PaulHoule• 11 years ago

I don't believe in the axiom of choice.

reagency10• 11 years ago

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.

tripzilch4• 11 years ago

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

arundelo• 11 years ago

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

pronoiac• 11 years ago

"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!"

VLM• 11 years ago

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.

tripzilch4• 11 years ago

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! ;-)

fmela• 11 years ago

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?

reagency10• 11 years ago

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.

black_knight5• 11 years ago

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.

JoshTriplett• 11 years ago

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

black_knight5• 11 years ago

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.

tel2• 11 years ago

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.

auggierose2• 11 years ago

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.

black_knight5• 11 years ago

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).

sjolsen• 11 years ago

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?

ColinWright8• 11 years ago

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 ...

tripzilch4• 11 years ago

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.

ebola17172• 11 years ago

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.

cmrx64• 11 years ago

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.

mikhailfranco• 11 years ago

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.

ColinWright8• 11 years ago

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.

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

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