The Banach-Tarski Paradox

[TSara]

Have I mentioned how much I dislike Math?

And I'm an Engineer.

[Mitchell]

All this thread has done is proved that that majority of us here will not be constructing a working Warp engine/core anytime soon. Myself included.

Dang My head hurts now to.

[Emberchyld]

Have I mentioned how much I dislike Math?

And I'm an Engineer.

Agreed Math is and evil necessity unless you're a math genius like Elessar...

[evcake]

^ Because if they're like all unhappy and stuff they can fall into like drugs and booze and stuff and they'll be like totally unemployed, get all frelled up and maybe even heaven forbid give up and jump off a cliff or something. THEN how are you gonna count. And then like in retrospect you'll be like oh I wish they'd got help before it was too late, etc etc. Better care now

I like this. All the numbers go off to sulk somewhere.

[CX]

Personally, I've never seen the point in doing mathematical proofs of things that are not physically possible. Since it's physically impossible for a sphere of infinite density to exist, then the Banach-Tarski Paradox has no application to reality and becomes a purely self-justified mental exercise.

It's like jogging on a treadmill. I get bored. What's the POINT? Where am I GOING?

Pretty much my view of it. Must be why I like my engineering classes that much more than the math ones even though they utilize essentially the same techniques. At least in engineering you're working toward something, like how much stress a member can take or how hot a jet turbine gets during combustion.

[Emberchyld]

Pretty much my view of it. Must be why I like my engineering classes that much more than the math ones even though they utilize essentially the same techniques. At least in engineering you're working toward something, like how much stress a member can take or how hot a jet turbine gets during combustion.

You're an ME, too, right?

That's why I like ME... very little theoretical and I can "touch" most of my work. EE classes made my head spin with things I couldn't see.

Still, I think I can see the draw of proving something like this paradox just because you can... I mean, I guess it's like trying to get a hole in one in (mini)golf-- it'll do absolutely nothing for you, but it's cool to say that you did it.

[Elessar]

Personally, I've never seen the point in doing mathematical proofs of things that are not physically possible. Since it's physically impossible for a sphere of infinite density to exist, then the Banach-Tarski Paradox has no application to reality and becomes a purely self-justified mental exercise.

It's like jogging on a treadmill. I get bored. What's the POINT? Where am I GOING?

Pretty much my view of it. Must be why I like my engineering classes that much more than the math ones even though they utilize essentially the same techniques. At least in engineering you're working toward something, like how much stress a member can take or how hot a jet turbine gets during combustion.

What most people don't realize though is that set theory forms the foundation for calculus, and of course you need calculus to develop differential equations, which govern the thermodynamics of fluids in a jet turbine. And abstract algebra (offshoot of set theory) is necessary to develop tensor analysis, from which the "stress tensor" is taken

MATH IS EVERYTHING!! hehehehe

Although the pure mathematics may seem like pure mathematics, they often lead to the application areas. The interesting thing about Math isn't that it's abstract and not real world applicable... it's that some areas of it describe Nature, and some areas of it (higher dimensional topology) are just purely theoretical...or so we think right now anyway

It actually turned out that topology (study of spatial geometry) was a purely theoretical area of math that nobody in physics or engineering needed bother themselves with until the early 20th century when someone came along and realized that differential geometry and topology and tensors DO describe something in nature... General Relativity!

So I guess that's why I love math... the things we use every day, are at their core, mathematical in nature, they're soooo far down there in the core that even the academic disciplines like aerospace engineering and electrical engineering that would seem to take advantage of a completely "more realistic" mathematics (like, algebra, calculus or diff eq) are actually using those abstract areas, it's just down at their foundation, where someone first invented those mathematics like 300 years ago by looking at THIS stuff and taking it a step further.

As far as this proof specifically, it can actually have a real world impact. I was reading up on the background of this and it was proposed in 1924 by, naturally, some dudes named Banach and Tarski. And originally the reason they put it forward was they did NOT agree with the Axiom of Choice (I explained the whole axiom of choice in another message, the one that didn't get posted somehow ). So their real intention was to produce this paradox and prove it was valid if we accepted the Axiom of Choice in order to say, "Look at how ridiculous this is if we accept the Axiom of Choice, this makes no sense, this is why the Axiom of Choice should not accepted." but Instead, for some reason historically, mathematicians came to accept the Axiom of Choice regardless, and instead, this paradox has come to represent the counterintuitive possibilities that exist as a result of it, and other axioms. There were, actually, paradoxes that existed as a result of some of the other, completely intuitive axioms, as well.

[Elessar]

My (second) attempt at a short and succinct explanation:

Axiom of Choice:

To begin with, an axiom is just a fundamental rule that you take for being true without arguments or evidence, it's something that's usually supposed to be logically intuitive.

The Axiom of Choice states that it is possible to construct a set of points, call it H, by taking exactly ONE point from a collection of sets, called C, regardless of how large C might be. This collection C can even be so large, that is it can contain so many subsets, that its number of subsets is called "uncountably large" which is just another way of saying "almost infinity". There's a more rigorous mathematical definition but it's not intuitive so it's not useful to explain here.

I added before that what distinguishes the Axiom of Choice is that there are 10 Set Axioms, and like all of them except the Axiom of Choice, were written like 400 years ago. The Axiom of Choice was proposed in 1904, so that's why it's rather unusual.

Explanation of Paradox:

The reason this allows this paradox to work is because this allows the "shape" and "volume" of the pieces you take apart the original sphere with, to be "undefined" and what's called "geometrically intractable", meaning you couldn't actually build them up physically, but they could exist. I was able to think of it as a spatial form of the Uncertainty Principle, whereby say some piece of information about a particle (its momentum, or energy or position) may or may not have some value, but you cannot measure it in quantum mechanics because it is an inextractable piece of information. The same could be said of geometry, shape, volume of the "pieces" you pick apart the sphere with, to construct the other two spheres.

The other reason that this works pretty easily is because the Real line (all non-complex numbers) on any number of dimensions (here we're working with 3D), is infinite. So there are an infinite number of points inside this sphere, and if you divide infinite by 2, you still get infinite, right? So when you reconstruct the other two spheres with the pieces you've taken apart, even if you have to divide the infinite number of points in the original sphere to do that, you still have an infinite number of points, and therefore the two spheres that result are STILL each identical to the original sphere.

The Axiom of Choice is what makes it possible to "construct" these pieces, or rather to just state indirectly that they CAN be constructed, without REALLY doing it. Because in the Axiom of Choice, you can have a set that is too large to count (uncountable), but still somehow construct that set H which contains exactly one point from all those uncountably large number of subsets. It's like saying that even if you have an infinite number of coconuts, you can still speak about a set which is made up of exactly ONE seed from each coconut, because even though you couldn't walk down a line of them and count a seed inside each one and ever finish (because they're again, "uncountable") you still KNOW there's a seed in each one, sort of based on a little leap of logic. That's why the Axiom of Choice was not always, and is still not Universally, accepted among mathematicians, because it allows, and requires, a bit of a "leap" from A to C by saying "we know those points are there, even if we can't look at them and count them."

These are the strange characteristics of the pieces of the original sphere which allow them to be divided up to form indeed 2, actually an INFINITE number of subsequent spheres of the same size.

Not Actually a Paradox:

So in reality it's not a paradox, it's only a paradox if you approach this "sphere" with the preconceptions that are associated with physical spheres, which are number 1, that the pieces of them are measurable, and number 2 that there are a finite number of pieces one can cut, whereas in a mathematical sphere there are infinite, and the pieces are immeasurable.

Also Why it Means Anything?

In a way, though, this has a sort of underlying metaphysical importance for how we approach mathematics and physics, too though. Because the difference in finiteness and infiniteness is manifested in the difference between our particle theories, and our field theories. Particle theories (nuclear, E&M) treat particles like electrons as point-like, with no discernible shape or size, , whereas the field theories deal with field lines and continuous structures. I am not far enough along in my education yet to really have a solid idea of this, but I am pretty sure there is a very important connection between the fact that there are "discrete" systems of mathematics (non-continuous, point-like environments) and continuous systems of mathematics, like calculus, and that there are also two respective ways that we approach modelling the physical Universe. The way we treat the "mathematical sphere" in this Paradox is the same way our modern picture of spacetime treats the fabric of space, as one continuous sheet of an infinite number of points - which is contradictory to what physical matter is like, with a finite number of atoms. As a lot of people know, there are efforts underway to prove that space is in fact distinct and discrete just like light and matter. That's part of the goal of string theory, is to discover the fundamental unit of measurement of physical space itself, to look for what would be little tick marks along the fabric of spacetime, a mininum amount of space that anything can move, which is almost silly to imagine but... You never know, Nature's done stranger things.

I just remembered another reason why this paradox and these "thought experiments" are important for science and engineering: The Navier-Stokes equation. The biggest challenge to CFD (computational fluid dynamics) right now, which i thought about getting into for graduate work because it's such a booming area for aerospace, is that the Navier-Stokes equation (which is a non-linear, non-homogeneous partial differential equation -- and that's just as scary as it sounds!) is intractable - it's insolvable. So, in order to really study what happens at extremely high-Reynolds number environments, which just means like high speed, high temperature, high altitude flight, they have to run all kinds of majorly complex computer simulations, because nobody can analytically solve the Navier-Stokes equation. It's actually in this booklet of one of the 10? mathematical problems of the 20th century that you can win a $1 million prize for solving. MoaningMinnie from House of Tucker, aka Species_1 from TrekBBS, is also a math student and told me she used to spend some time every week trying to solve it, hehe. But developing new Axioms or debunking old ones, is basically the only way to reveal a previously-unseen path to an analytical solution of the Navier-Stokes equation, which would have untold benefits for the aerospace and mechanical engineering industries, revolutionizing the way we understand high altitude fluid dynamics. And probably putting a lot of CFD students out of a job

[evcake]

Thanks for trying, though.

[Kevin Thomas Riley]

This is me, thinking about math and remembering math classes in our equivalent of high school -->

[JadziaKathryn]

Aha! I got the coconut part. See, "infinite sets" and such I can't understand. But infinite coconuts I can imagine. I sort of get this, in a rough sense, but you lost me at the part where this is supposed to be useful, except where one could win a million dollars. That would be useful.