
The Point of the Banach-Tarski Theorem – not just a curiosity - ColinWright
http://www.solipsys.co.uk/new/ThePointOfTheBanachTarskiTheorem.html?HN_20150607
======
murbard2
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.

~~~
j2kun
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...](http://mathoverflow.net/questions/129036/counterintuitive-consequences-
of-the-axiom-of-determinacy)

~~~
reagency
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.

~~~
black_knight
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.

------
arundelo
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](http://www.irregularwebcomic.net/2339.html)

~~~
pronoiac
"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!"

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

~~~
tel
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.

------
mikhailfranco
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.

~~~
ColinWright
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.

------
sjolsen
>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?

~~~
ColinWright
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 ...

~~~
tripzilch
> 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.

------
gus_massa
> _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.

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

------
summerdown2
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
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.

~~~
summerdown2
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.

~~~
ColinWright
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.

~~~
summerdown2
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?

~~~
ColinWright
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?

~~~
summerdown2
> 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?

~~~
reagency
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.

~~~
SamReidHughes
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.

------
legulere
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
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.

~~~
reagency
But uncountable sets do NOT model reality well.

~~~
digama
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.

------
cmrx64
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.

------
fmela
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?

~~~
reagency
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.

------
VLM
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...](http://en.wikipedia.org/wiki/Non-
measurable_set#Consistent_definitions_of_measure_and_probability)

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.

~~~
tripzilch
> 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! ;-)

------
PaulHoule
I don't believe in the axiom of choice.

~~~
reagency
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.

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

