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Layman's Guide to the Banach-Tarski Paradox

By The Writer in Science
Sat May 24, 2003 at 02:08:57 AM EST
Tags: Science (all tags)
Science

The Banach-Tarski Paradox is well-known among mathematicians, particularly among set theorists.1 The paradox states that it is possible to take a solid sphere (a "ball"), cut it up into a finite number of pieces, rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere. In other words, you've doubled the volume of the original sphere.

"Impossible!" I hear you say. "That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox. But I'd like to point out in this article why this may not be as impossible as one might think at first.

1It revolves around the decades-old debate of whether the Axiom of Choice should be admitted or rejected. More on this in the epilogue.


Preliminaries

First of all, let's nail down what exactly we're talking about so that we're all on the same page.

First and foremost, we're talking about a mathematical sphere, not a physical sphere, although I'd like to use an analogy with physical spheres to describe one possible way to intuit the Banach-Tarski Paradox. By mathematical sphere, I mean the set of points that lie within a 3-dimensional spherical area in ℜ3, where ℜ is the set of all real numbers. For simplicity, let's assume a radius of 1, so our sphere would be the set:

S = {(x,y,z) | x2+y2+z2 <= 1 }

One important difference between S and a real, physical sphere is that S is infinitely divisible. Mathematically speaking, S contains an infinite number2 of points. This is not true of a physical sphere, as there are a finite number of atoms in any given physical sphere; so a physical sphere is not infinitely divisible.

2Or, to be precise, c points, where c is the cardinality of the continuum.

In fact, if we assume that spheres are not infinitely divisible, then the Banach-Tarski paradox doesn't apply, because each of the "pieces" in the paradox is so infinitely complex that they are not "measurable" (in human language, they do not have a well-defined volume; it is impossible to measure their volume). Immeasurable pieces can only exist if the sphere can be cut into infinitely-detailed pieces; this obviously isn't true for real spheres, since you cannot cut atoms into arbitrary shapes, especially not into infinitely complex shapes.

Now let's move on to the paradox itself.

The Banach-Tarski Paradox

The Banach-Tarski Paradox states, basically, that it is possible to take S (as we've defined above), and cut it up into n disjoint pieces, which we shall call A1, A2, ... An, whose union is S itself, where n is a finite number, such that if we perform some (finite) sequence of rotations and translations on each of the Ai's, we will end up with two copies of S. Or, to be mathematically precise:

  • Ai ∩ Aj = ∅ for each i and j between 1 and n such that i≠j (no two pieces overlap each other)
  • A1 ∪ A2 ∪ ... ∪ An = S (assembling all the pieces yields the original sphere S)
  • There exist T1, T2, ... Tn, where each Ti represents some finite sequence of rotations and translations, such that if we apply each Ti to each Ai (let's call the result Ai'), then:
    • A1' ∪ A2' ∪ ... ∪ Am' = S (a subset of the original pieces forms S)
    • Am+1' ∪ Am+2' ∪ ... An = S' (the remaining pieces forms a copy of S)
    where m is some number between 1 and n, and S' is S translated by some finite amount so that S and S' are disjoint.

It is interesting to note that one corollary of this paradox is that you can take a sphere, cut it into n pieces, remove some of the pieces, and reassemble the remaining pieces back into the original sphere without missing anything. Obviously, this is impossible with a physical sphere; but it is quite possible with mathematical spheres (which are infinitely divisible), if the Axiom of Choice is assumed.

(Before you dismiss this notion outright, let me state that mathematically infinite objects do not always behave intuitively. As a comparison, we use a more intuitive example of duplicating the set of integers: given N, the (infinitely large) set containing all the integers, we can split them up into two sets, E containing all the even integers, and F containing all the odd integers. Are E and F each smaller than N, the set of all integers? Intuitively, it appears to be so; however, I will convince you that they are, in fact, the same size. First, we take E, and rename each member of E so that a number x is renamed to x divided by two. What do we get? We now find that E=N. Similarly, we take each member y from F, and rename y to (y-1)/2. Whoopie, we also find that F=N. We have just duplicated the set of integers using nothing more than just the original integers. We didn't even need to use infinitely-divisible freak objects to achieve this.)

Coming back to spheres, it is helpful to keep in mind that each of these pieces Ai are potentially infinitely complex so that they do not have any well-defined volume. Now, this whole paradox may seem remotely possible if we had, say, required 1,000,000 pieces to achieve our feat; there's intuitively more room for a sleight of hand if we had a million pieces to play with. However, the kicker about this whole paradox is that we don't need more than five pieces to achieve this feat. And unlike our odd/even number example, we do not need to play tricks with renaming individual points of each piece; we can perform the miracle by merely using well-behaved operations like rotations and translations. Furthermore, one of these pieces only needs to contain the single point at the centre of the sphere.

In other words, it is mathematically possible to cut S into a mere four pieces (if we disregard the one center point), and to reassemble two of these pieces into the original S, and reassemble the other two into a copy of S.

The catch, of course, is that each of these four pieces are so complex that they do not have any "measure" (i.e., their respective volumes are not well-defined), and that we do not know how to mathematically describe them other than the fact that they exist and exhibit the strange re-assembly property. In fact, it is quite possible that each of those pieces consists of isolated points spread out throughout the entire volume of the original sphere S.

How can this ever be intuitive??!

I promised to convince you that this bizarre mathematical phenomenon isn't all that strange, after all. So here's my proposed "intuitive" rationalization of it. I'll do it by way of an analogy with a physical sphere.

Let's forget for the moment the mathematical sphere S, which has infinite density. Let's consider a real, physical sphere B (for "ball"), also of radius 1. B is identical to S except that it consists of a finite (albeit large) number of atoms. The way these atoms are laid out in B is called the crystalline structure of B. (I.e., if you take B, or any physical object for that matter, and look at it under an electron microscope, you will see the atoms laid out in a fixed, regular pattern. That's called its "crystal lattice".) Usually, the crystalline structure is a simple geometric relationship between neighbouring atoms.

Notice that although the geometric relationship between atoms define its crystalline structure, the precise distance between atoms may vary. This leads to materials of different densities.

Now, we perform the equivalent of a Banach-Tarski decomposition on our physical sphere B: we "atomize" B into four spherical clouds of atoms, let's call them C1, C2, C3, and C4. (We'll ignore the central atom in B, just as in the mathematical version of this decomposition.) Let's assume that each of these clouds are sparse enough that they are gaseous, no longer solid by themselves (imitating the immeasurability of the mathematical pieces of S). Furthermore, let's say that the atoms in each of these clouds are laid out in a regular pattern, so that if we rotate C1 by some angle G, and put it together with C2 in the same spherical region, the atoms in both clouds line up into the same crystalline structure as B, except that now the distance between atoms is greater (to account for the missing atoms now in clouds C3 and C4). Similarly, assume we can do the same with C3 and C4: we just translate them away from the original spherical region of B so that they don't interfere with C1 and C2, and reassemble them into another sphere.

Now, we have successfully built two (physical!) spheres with the same radius as B, using only material from B itself. Each of the two spheres have the same crystalline structure as B. The only difference between these spheres and B is that they each have only half the density of B.

To bring this analogy back to the mathematical sphere S: we can think of the infinitely complex pieces A1, ... A4 as the equivalent of "atom clouds", which are non-solid (immeasurable). Think of the crystalline structure of S as the topological structure of points in ℜ3. These "clouds" lack this "crystalline structure" (i.e., they are unmeasurable); but by suitable rearrangement of them, we can form them into two identical spheres, with half the density of the original, so that they do have the same "crystalline structure" (i.e., the resultant two spheres are well-behaved, measurable sets). These two spheres are identical to S, except for having only half the density of S. However, S is infinitely dense, and so are its pieces A1, ... A4. This means the two resultant spheres are still infinitely dense. That is to say, they are identical to S.

Bingo! There is no paradox here after all. We are merely seeing the logical consequence of mathematical sets like S being infinitely dense. In fact, if you think about it, this is not any stranger than how we managed to duplicate the set of all integers, by splitting it up into two halves, and renaming the members in each half so they each become identical to the original set again. It is only logical that we can continually extract more volume out of an infinitely dense, mathematical sphere S.

Epilogue

Now, having convinced you that the Banach-Tarski Paradox isn't really that strange after all, I'd like to mention that the derivation of this paradox depends on the Axiom of Choice, and although most mathematicians accept the Axiom of Choice, not all agree with it. There has been much debate over the merit of adopting this axiom, as well as research into the consequences of choosing either way: it does simplify a lot of mathematical proofs, but it also introduces strange results like the Banach-Tarski paradox which we just discussed.

If you're unfamiliar with the Axiom of Choice, it basically goes like this: if you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H.

In the case of the Banach-Tarski paradox, each of the infinitely complex "pieces" of the sphere S is built from these choice sets. Since we do not know of any algorithm to actually construct these sets, we can only indirectly infer some of the properties of the "pieces", such as their not having a (Lebesgue-) measure (i.e., they have intractible geometric complexity). Some of the debate surrounding the Axiom of Choice revolves around whether these non-constructible sets are mathematically admissible. The reader is encouraged to make good use to Google for more information about this debate; it is too vast a topic to explore in this article. It suffices to say that most mathematicians adopt the Axiom of Choice, simply because of the usefulness of results that can be derived.

One might wonder, then, about what would happen if we didn't assume the Axiom of Choice. We do know that we would likely be unable to derive the Banach-Tarski paradox; however, we also know that paradoxical sets do exist even without the Axiom of Choice. These paradoxical sets exhibit the same "weird" behaviour of the Banach-Tarski spheres, in that you can decompose these sets into a finite number of parts, and reassemble them into multiple copies of the original.

For more information, here is more information about the Axiom of Choice. Also check out the funny animation on this page about the Banach-Tarski Paradox.

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Poll
Is this a plausible rationalization of the Banach-Tarski Paradox?
o Yes! Makes sense to me! 36%
o Yes, it sound plausible but I really have no idea 16%
o No, it sounds ridiculous, even though I still have no idea 2%
o No! The Axiom of Choice cannot be true! Argghrhghrgh 5%
o Banach and Tarski should be institutionalized, whoever they are 15%
o I am pro-Axiom of Choice. 23%

Votes: 96
Results | Other Polls

Related Links
o Google
o Axiom of Choice
o crystallin e structure
o more information about the Axiom of Choice
o Banach-Tar ski Paradox
o Also by The Writer


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Layman's Guide to the Banach-Tarski Paradox | 417 comments (354 topical, 63 editorial, 1 hidden)
Easy answer (2.60 / 10) (#5)
by marcos on Fri May 23, 2003 at 01:57:32 PM EST

If 9.999999.... = 10, and infinite/2 = infinity, then every object can be split into two objects that are the size of the same object.

The fact of the matter is - there is no such thing as infinity. It does not exist in our physical world, and if you wish to play with a theoretical and undefined infinity, then you have got to neccesarily not compare those objects with natural objects.

No no (none / 0) (#10)
by melia on Fri May 23, 2003 at 02:08:12 PM EST

His explanation was incredibly helpful, and I would very likely not have clearly understood without the comparison. Arguments about the nature of infinity I shall leave to others more qualified.
Disclaimer: All of the above is probably wrong
[ Parent ]
That is the entire point (none / 0) (#15)
by marcos on Fri May 23, 2003 at 02:28:49 PM EST

There is a deliberate obfuscation when mathematics goes into infinity that allows such things to happen. It cannot happen.

[ Parent ]
So... (none / 0) (#16)
by melia on Fri May 23, 2003 at 02:35:45 PM EST

...i can't be introduced to the mathematical concept of infinity because introducing infinity will obfuscate the concept? I don't understand you, you're confusing me!
Disclaimer: All of the above is probably wrong
[ Parent ]
no... (4.33 / 3) (#113)
by laotic on Sat May 24, 2003 at 04:19:38 AM EST

...I think that what he's saying is that you can happily enjoy being introduced to the mathematical concept of infinity and wallow in it as long as it feels pleasant, but that as soon as you start drawing comparisons with our real (perceived) world, you cannot apply it much.

Sig? Sigh.
[ Parent ]
Actually, you're right (5.00 / 1) (#28)
by The Writer on Fri May 23, 2003 at 03:31:14 PM EST

Although the Banach-Tarski paradox talks only about spheres and actually requires the Axiom of Choice, mathematicians have proven (without using the Axiom of Choice) that every n-dimensional object (including 3-dimensional objects) with finite extent can be decomposed into finite number of pieces and reassembled into any other n-dimensional object with finite extent. The resulting object does not have to be equal or smaller in size from the original; in fact, it is theoretically possible to cut a (mathematical) sphere the size of a pea into a finite number of pieces, and reassemble these pieces into a sphere the size of the Sun.

As far as comparisons with the physical world is concerned, it is mainly just to aid one's mental conception of the mathematical object; since we are most familiar with the behaviour of the physical world. But just because something is not physical does not necessarily mean it cannot exist (unless, of course, you define existence to be only physical existence). But that gets into the philosophy of mathematics, and I've no wish to go there at the moment.

And contrary to what you claim, mathematical infinity, or should I say, infinities, is very well-defined. Infinities are rigorously derived from the axioms of set theory (i.e., no such nonsense as the high-school math explanation of "now imagine if you repeat this infinitely many times"), and their properties can be rigorously derived from the same axioms. They may not correspond to anything in the physical world; but they are certainly very well defined, and aren't some vague carpet which we conveniently sweep undesirable details under.

[ Parent ]

Set theory (3.50 / 2) (#50)
by marcos on Fri May 23, 2003 at 04:36:09 PM EST

I'm not sure what you are talking about, but do you mean the various things we learnt in Algebra like Körper and Kommutativer Ring? Sorry, I don't know the english words, never learnt em in English.

If you are talking of those definations of sets, which specify the properties of sets, all they do is define the properties of the sets, and assume that all elements with those properties can be as many as  are needed, implying infinity.

The very defination of existence in our physical world implies a lower limit. If there is a lower limit, then there is neccesarily an upper limit, since the upper limit is composed of stacks of objects of the size of the object at the lowest limit. So, I speak as an engineer, and not a mathematician, since every object has got to construct itself or be constructed in finite time, of finite pieces of material, it neccesarily cannot stretch into infinity.

Infinity is defined, but its very defination has a  flaw, and this lies in the statement

infinity / 2 = infinity. Obviously, after performing that operation, you have got two objects of different sizes, and if you fail to take this into consideration, then you can do lots of whack shit that will never ever translate into reality.

The paradox is paradoxical, because we try to understand it in terms of our physical world. But if you do so, then you have got to accept that there is no infinity, in which case the paradox vanishes.

You talk twice if finite:

finite extent, decomposed into finite pieces.

but you don't say

decomposed into finite pieces, by an infinite number of cuts.

You add that, and you are using infinity again, which, according to my senses, fudge things up again. Astound me by having finite and closed sets everywhere in your calculations.

[ Parent ]

Wait a second here (5.00 / 3) (#61)
by The Writer on Fri May 23, 2003 at 05:30:46 PM EST

I think you're confusing physical reality with mathematical definitions.

I completely concede that in the physical world, there is no such thing as a completed infinity. At least, not as far as we know today. You won't find Banach-Tarski decompositions in the physical world (or at least in the macroscopic world) anytime soon; neither will you find a physical structure equivalent to the set of natural numbers that contains an infinite number of objects.

However, that doesn't mean infinite things aren't well defined in mathematics. Just like you said, sets are defined by their properties, and we can have as many elements as it takes to satisfy those properties. In this sense, we have well-defined, and consistent theorems about the behaviour of infinite objects, at least within the realm of mathematics.

My whole purpose of using a physical analogy is simply to provide a basis on which the reader can generalize to the (arguably hypothetical) infinite realm of mathematics, and have some intuition about why it would behave the way predicted by mathematical theorems. I wasn't trying to say that the physical sphere behaves the same way as the mathematical sphere --- the existence of the paradox shows that they don't behave the same way.

Now on another note, be careful before dismissing anything that contains infinite quantities as having no correlation with the "real world". We should always keep in mind that our perception of the real world is shaped by the behaviour of objects in the macroscopic scale, between the subatomic scale and the astronomical scale. We make generalizations about the behaviour of the universe based on this limited scale of perception; however, as General Relativity and Quantum Theory shows us, when we move to the extreme ends of the spectrum, the physical universe behaves in rather different ways from what we might expect.

Some examples from physics: when calculating magnetic fields from electric fields, the calculations involve imaginary numbers (or more appropriately, complex numbers). When mathematicians first came up with complex numbers, people scoffed at them. There is no such thing as a square root of -1, what are you talking about?! However, later we find out that the theory of electromagnetism involves calculations using complex numbers.

Another example: one of the breakthroughs in theoretical physics of this century was the so-called electro-weak theory, which unifies electromagnetism with the nuclear "weak force". This theory is quite applicable today; it is the basis for calculating the behaviour of particles in nuclear reactions. Now, when this theory first came out, people had a strong objection to it. The problem was that although the equations balanced, the quantities on each side of the equation were infinite; only, they have a finite difference so the results are always finite.

The physicists who formulated the theory simply used a normalization process by "cancelling out" the infinite terms from each side of the equation. For a long time, people could not accept this seemingly-illegal operation. However, as far as we can tell today, the predictions of the electroweak theory are completely consistent with the observed behaviour of subatomic particles. The results that came out of that equation that contained infinite quantities is actually consistent with the physical world.

None of this is "intuitive", but the results are borne out through experimentation. So one should be careful about dismissing mathematical results outright just because they are non-intuitive.

And on the Banach-Tarksi paradox, I read an article some time ago that some physicists are actually finding out that the bizarre sphere-reassembly process is actually applicable to some processes in high-energy particle physics. I don't remember the details, unfortunately; but it shows you that math is not to be dismissed lightly.

[ Parent ]

Yup (none / 0) (#74)
by tetsuwan on Fri May 23, 2003 at 07:21:50 PM EST

I remember t'Hooft's Nobel lecture, although it disappointed me at some points where it left out the mathematical candy.

Njal's Saga: Just like Romeo & Juliet without the romance
[ Parent ]

The hardest button to button... (none / 0) (#153)
by Xenex on Sat May 24, 2003 at 12:52:27 PM EST

Putting a Whit Stripes lyric in your .sig has been an effective way to make me remember you.

And it's driving me mad!

It's what's not there that makes what's there what it is.
[ Parent ]

Why is it driving you mad? [n/t] (none / 0) (#213)
by tetsuwan on Sun May 25, 2003 at 07:45:44 AM EST


Njal's Saga: Just like Romeo & Juliet without the romance
[ Parent ]

Well... then I think of the song... (none / 0) (#268)
by Xenex on Sun May 25, 2003 at 06:00:58 PM EST

Which leads to playing the album again... which I guess isn't necessarily a bad thing...  

It's what's not there that makes what's there what it is.
[ Parent ]
Complex Numbers and Physics (5.00 / 3) (#187)
by Neil Rubin on Sat May 24, 2003 at 10:25:25 PM EST

Some examples from physics: when calculating magnetic fields from electric fields, the calculations involve imaginary numbers (or more appropriately, complex numbers). When mathematicians first came up with complex numbers, people scoffed at them. There is no such thing as a square root of -1, what are you talking about?! However, later we find out that the theory of electromagnetism involves calculations using complex numbers.

It is certainly true that complex numbers are used to solve problems in electromagnetism, but they are actually no more than a notational convenience. Really, they are just being used to describe either the rotation of vectors in two-dimensions or (the closely-related case of) sinusoidal oscillations. If you look at a text on the subject, you will find the use of certain operations:

  • complex norm (a.k.a. vector length)
  • multiplication by e (a.k.a. vector rotation)
  • complex addition (a.k.a. vector addition)
  • possibly something like Re(z1*z2) (a.k.a. vector dot-product)
You never see anything that you can't do with simple 2-D vector like generic complex multiplication. The physical laws being described are invariant under rotations, so that taking the complex number interpretation too seriously would cause big problems. Why should squaring one of the basis vectors give +1 and squaring the other give -1? Even if you represent electric fields with real numbers and magnetic fields with imaginary ones, the problem remains, since Maxwell's equations (in free space) are invariant under rotations between electric and magnetic fields, the so-called duality transformation.

There are places in quantum mechanics where more detailed properties of the complex numbers are used. If you've taken a bit of quantum mechanics, you might think that the wavefunction is an example, but again the complex numbers are just being used to describe the rotations of 2-D vectors. The absolute phase of a wavefunction is unmeasurable and any physical system is invariant under the simultaneous rotation of all wavefunctions. One the other hand, the detailed properties of analytic functions (basically, these are functions of complex variables for which derivatives exist with respect to those variables. They are the main topic of any course called "Complex Analysis".) are extremely useful. I don't know how to go about proving many essential results without appealing to their existence.

[ Parent ]

the world is full of infinities (5.00 / 2) (#100)
by martingale on Fri May 23, 2003 at 11:26:20 PM EST

Your judgement is clouded by a narrow focus on the concept of an infinitely large number, but if you think about it some more, you'll see that the world is full of opportunities for infinite sets.

Do you believe in the number two? A nice, finite number, no infinities there. Now what's two? two equals one plus half of two. A simple, finite equation. Also, two equals one plus one half of (one plus one half of two). Another simple, finite equation. How many equations exist, in the real world, of this type? Is any of those equations less real than the first one I've given? But the world now contains an infinite series of equivalent objects.

Now you might say this is contrived. But is it? I've only written down an equilibrium condition. Most dynamical systems in engineering are intimately related to equilibria. Each equilibrium equation generates an infinite number of equivalent versions, each slightly more complex than the preceding one.

So now we have an infinite number of completely equivalent descriptions of a real world system. How do you deny the existence of such a concrete, but infinite, list of things? Which one of these equations is not real enough so that it doesn't apply to the real world (assuming at least one of them "applies") ?

Infinity is everywhere you can think.

[ Parent ]

Translations (5.00 / 2) (#107)
by Hideyoshi on Sat May 24, 2003 at 01:54:21 AM EST

I'm not sure what you are talking about, but do you mean the various things we learnt in Algebra like Körper and Kommutativer Ring? Sorry, I don't know the english words, never learnt em in English.
"Fields" and "Commutative Rings".

[ Parent ]
"existence" of infinity (5.00 / 2) (#121)
by kmlee99 on Sat May 24, 2003 at 06:38:44 AM EST

If 9.999999.... = 10, and infinite/2 = infinity, then every object can be split into two objects that are the size of the same object.

What do you exactly mean by "...." in 9.999999.... ? Do you have a proper definition of the division operator "/" for the object infinity (which is not a real number)?

The fact of the matter is - there is no such thing as infinity. It does not exist in our physical world,

But neither do the number like 2 or log(pi). Infinity, like numbers, is an abstraction which has proven its usefulness in the real world. Like most mathematical objects, whether it "exists" physically is not very relevant.

and if you wish to play with a theoretical and undefined infinity, then you have got to neccesarily not compare those objects with natural objects.

There is no problem at all if you just use them in limit operations, e.g. lim n->infinity (n/2) = infinity, rather than infinity / 2 = infinity.

[ Parent ]

Can't resist feeding the troll (none / 0) (#126)
by dcturner on Sat May 24, 2003 at 08:28:00 AM EST

As you disagree with 9.99999... = 10 I can deduce that you think at least one of the objects mentioned exists. I'll assume you think that 10 exists. Then I guess you think that 1, 2, ..., 9 all exist too. And addition? So 11 exists. 12 exists. Where do things stop existing? I mean, 47 is pretty large but someone once told me that 53 existed which blows that idea out of the water.

And since when has infinity been undefined? The story says there are c points in the sphere, where c can be well-defined as the class of sets which biject into the first uncountable ordinal. The first uncountable ordinal can be defined as the set of countable ordinals. The countable ordinals can be constructed from the finite ordinals and they can be constructed from 0. Does 0 exist?

Remove the opinion on spam to reply.


[ Parent ]
Troll indeed! (none / 0) (#129)
by marcos on Sat May 24, 2003 at 08:52:48 AM EST

Look, just because you cannot perform math well does not mean that anyone who says something you do not understand is a troll.

A sphere is a mathematical object, and it is a physical object. The entire paradox arises because we try to imagine the sphere as a physical object. But a physical object is not composed of an infinite number of particles, hence this paradox is not paradoxical in the real world. That was my point.

Your supposed sarcasm is just irritating, please don't do it again.

Infinity is defined mathematically, since the sets we are working with are not closed, but can always gain new objects of the same property. Say for example, we have a set with the mathematical property defined. If there exists and x, and there exists and x+1 for all objects, then an infinite number of objects can fit into the set obviously. However, the set of atoms, or whatever makes up a sphere is not defined that way.

A mathematical sphere is composed of a set of numbers. A physical sphere is composed of a set of  atoms. Different things, and for one, infinity exists, and for the other it does not.

[ Parent ]

It's not a troll (5.00 / 1) (#274)
by scheme on Sun May 25, 2003 at 09:46:32 PM EST

Infinity is defined mathematically, since the sets we are working with are not closed, but can always gain new objects of the same property.

Actually, the sets in question R^3 are closed under addition and multiplication. They have to be in order to form a field. True they aren't closed under algebraic operations (You need to go to C^3 to do so).

However, this illustrates an important point in mathematics. You really need to be precise about your terminology and what words you use to define things. To do otherwise leads to errors and misunderstandings.

Going back to your original post, infinity/2 does not necessarily equal infinity. Also infinity is not an undefined thing, it is a fairly well defined object due to the work of many mathematicians.

I have to agree with the poster above, you labelled you as a troll. Especially in light of your other posts equating GR and QM. Either you are glossing over things and intentionally posting misleading statements or your understanding is a bit lacking.


"Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT'S relativity." --Albert Einstein


[ Parent ]
What about... (none / 0) (#131)
by CodeWright on Sat May 24, 2003 at 09:16:55 AM EST

Singularities?

Past the Schwarzchild radius, matter falling into a singularity takes an infinite amount of time to hit bottom, constantly accelerating along the way.

In fact, this Banach-Tarski paradox seems like it might very well have a real world application with regard to gravitational singularities.

If you could conceivably split a given black hole in two, each half would still remain infinitely small, infinitely dense, and infinitely deep (although the Schwarzchild radius of each would probably be smaller).

--
"Jumpin Jesus H. Christ riding a segway with a little fruity 1 pint bucket of Ben and Jerry's rainbow fairy-berry crunch in his hand." --[ Parent ]
No, you are wrong (1.50 / 4) (#139)
by marcos on Sat May 24, 2003 at 10:04:58 AM EST

You know why? Einstein told us E=mc². You know what that means? It means that mass can be represented as E/c², implying that the minimum possible size of mass is 1/c², where 1 represents a single unit of energy. Energy cannot be created, so a single piece of energy does not have any constituent parts. There is no 1/2 energy.

As such, a real singularity is not actually singular, when we look at its energy. The more mass it consumes, the more energy it gains, and the more mass it has as a result.

If you divide a black hole, each black hole has got a different energy level, implying that they are smaller, energywise as well as matter wise.

On a macroscopic level, there is no difference, since both very small to us, but on an energy level, two very different sizes.

In any case, black holes are theoretical occurences, and with that amount of matter all in one place, I believe that the formulae are too simplified to accurately predict what happens when matter collapses into itself under the force of its own gravity.

This is not proven stuff, so till we see it proven practically as well as theoretically, I would accept it. In any case, the core point is that a black hole is not infinitely small, it is just small. It is not a mathematical singularity, it is a physical singularity, which again, are very different things.

[ Parent ]

Huh?? (5.00 / 3) (#178)
by The Writer on Sat May 24, 2003 at 06:38:59 PM EST

Can you please explain how you got from E=mc2 to concluding that the smallest unit of energy is 1/c2? It seems like you're jumping to conclusions without accounting for the units on either side of the equation.

Also, General Relativity does not have any lower bound on the amount of energy one can have; that is quantum theory. Even today, physicists are still trying to reconcile General Relativity with quantum theory. Unless you are in line for the Nobel Prize for a yet unannounced theoretical breakthrough, I will have to be very skeptical of what you just claimed.

And as for the blackhole not being infinitely small, well, you may be right, but until General Relativity is reconciled with quantum theory, we have no choice but to conclude that once a mass starts collapsing under its own weight, there is no limit to how far it can collapse, and therefore, a singularity is by necessity infinitely small. At least, this is what the equations tell us. (However, from the perspective of a particle in the blackhole, space would still be as large as before, and hence non-zero. Such is the nature of relativity.)

[ Parent ]

I am in line for the nobel prize (none / 0) (#198)
by marcos on Sun May 25, 2003 at 03:52:56 AM EST

Quantum theory and relativity are the same thing said in different ways. They both equate energy and mass - one says energy only exists in quantums, the other that energy level is proportional to mass. In both cases, mass and energy are made interchangeable. In neither case can energy be split. In neither case can energy be made smaller than it is. It follows that in neither case can energy/mass stacked on top of itself be like a mathematical singularity.

[ Parent ]
Gross simplification (none / 0) (#261)
by discoflamingo13 on Sun May 25, 2003 at 04:26:18 PM EST

Infinity is not a number - infinity is a process. Infinity is about what we consider the limits of extremely long or precise processes - you would not agree that 9.99999999999999999999999999 is 10, but the process of taking progessively closer measurements (in steps of 9/(10*n) increments) will eventually reach 10, yes?

It is through the process of mathematical analysis (the properties of sequences, series, and functions on the real and complex sets) that allow us to see how calculus can be considered accurate on a theoretical level. For about 99.9% of all functions, derivatives, and integrals you will encounter in the "real world", calculus will work perfectly well. But to prove to ourselves that it works logically, the concept of limits and infinity needs to be introduced.



The more I watch, the more I learn ---
If you set yourself on fire, the world will pay to watch you burn.
--- Course of Empire

[ Parent ]
The root of the 9.999..=10 (5.00 / 1) (#353)
by JonesBoy on Tue May 27, 2003 at 11:34:55 AM EST

He is referring to this

a=9.99999... (repeating forever)

 10*a=99.9999...
   -a=-9.9999...
==================
  9*a=90
therefore, a=10

(excuse the ascii art)
Speeding never killed anyone. Stopping did.
[ Parent ]

+1 FP - I concur with all these findings (3.58 / 12) (#6)
by Tex Bigballs on Fri May 23, 2003 at 01:57:45 PM EST

it's good to finally see other super geniuses like myself on this site.

This one was easy: (none / 0) (#30)
by tetsuwan on Fri May 23, 2003 at 03:35:38 PM EST

"Catch me if you can" (I haven't even seen it)

Njal's Saga: Just like Romeo & Juliet without the romance
[ Parent ]

Explanation of notation (4.80 / 5) (#11)
by The Writer on Fri May 23, 2003 at 02:11:46 PM EST

(By request)

The symbols ∩ and ∪ stand for set intersection and set union, respectively. A mathematical "set", roughly speaking, is a collection of (mathematical) objects. We say x∈S if x is an element of, or a member of, the set S. (We say that S "contains" x.)

The intersection of two sets X and Y, written as X∩Y, is simply the set where every element x is both in X and in Y; i.e., x∈X AND x∈Y. Or, in other words, X∩Y is the set that contains all elements common to both X and Y.

The union of two sets X and Y, written as X∪Y, is simply the set where every element x is either in X or in Y. I.e., x∈X OR x∈Y (or both).

There is something called the empty set, written as ∅. This is the set that does not contain anything. In the context of this article, if the intersection of any two or more sets is ∅, that means there are no common elements between those sets.

Thanks, I needed to feel stupid today. (4.00 / 1) (#17)
by Kasreyn on Fri May 23, 2003 at 02:35:48 PM EST

Trig and basic physics were as far as I went into math, and I forgot those years ago (except for the quadratic equation). This article made my head hurt.

Abstaining because I'm totally unqualified to judge this one on its merits. I'm going to go curl up in the fetal position now...


-Kasreyn


"Extenuating circumstance to be mentioned on Judgement Day:
We never asked to be born in the first place."

R.I.P. Kurt. You will be missed.
A real, self-contained proof (4.83 / 6) (#20)
by i on Fri May 23, 2003 at 02:45:25 PM EST

can be found here [pdf]. Some understanding of abstract algebra, measure theory and analysis required.

and we have a contradicton according to our assumptions and the factor theorem

No, I'm afraid you're mistaken. (2.00 / 12) (#21)
by Hide The Hamster on Fri May 23, 2003 at 02:48:40 PM EST

I'm sorry. Regretfully, I must take your junior mathematician badge away.


Free spirits are a liability.

August 8, 2004: "it certainly is" and I had engaged in a homosexual tryst.

uh...yeah, that thing! [nt] (3.00 / 1) (#25)
by VoxLobster on Fri May 23, 2003 at 03:20:36 PM EST


VoxLobster
I was raised by a cup of coffee! -- Homsar

Any mathematical theorem... (1.95 / 22) (#26)
by Random Number Generator Troll on Fri May 23, 2003 at 03:20:44 PM EST

...that involves the word infinity is inherently rubbish.

What is it with all these people getting crappy theorems named after them? Fucking Dirac Pulses for example. Intimidating name, but is it difficult? Does Dirac really deserve that named after him? Does he fuck!

As far as I can tell, the banal theorem explained in this article is just an extension of the stupefyingly banal 'what is infinity divided by x?' question.

To make myself feel better, and to get my name in the dictionary, here is the RNGT Theorem:

x + ( y * 2 )
------------- = x
(where x = y)
3


The world of mathematics shall forever quiver in my shadow.

Dividing infinity (5.00 / 1) (#57)
by jonathan_ingram on Fri May 23, 2003 at 05:22:09 PM EST

You want to do arithmetic with infinity? Let me introduce you to surreal numbers (mentioned in an old Kuro5hin story).
-- Jon
[ Parent ]
Infinite arithmetic (5.00 / 2) (#63)
by The Writer on Fri May 23, 2003 at 05:44:04 PM EST

Or for that matter, do a google search on "hyperreals".

Hyperreals are an extension of real numbers, that are fully closed under addition, subtraction, multiplication, and division. Moreover, they include infinitesimals (numbers which are infinitely small yet non-zero---calculus buffs will recall how these quantities are used in the definition of the derivative, but are skirted around using limits in conventional calculus texts) and infinite reals. I found hyperreals very interesting, since Cantor's transfinite numbers are discrete, but this one has a continuum of infinite numbers.

[ Parent ]

Such crappy theorems have made the world as is (5.00 / 3) (#150)
by svampa on Sat May 24, 2003 at 12:24:46 PM EST

In the background of mathematic used for buildings, rockets, space crafts, computers... is infinity.

Do you think simple formulas that engeneers use have been invented as you can see them now?. No, they are the final issue of complex theorems that envolve concepts like infinity. Try to find any physics that doesn't comes from integrals and derivatives.

Perhaps weird theorems look absurd now, but a lot of current techology were weirds theorems that someone found could be applicable to real world.



[ Parent ]
Simple formulas (5.00 / 2) (#174)
by The Writer on Sat May 24, 2003 at 06:20:45 PM EST

Right on!

A lot of simple-looking formulas and equations are the result of mind-numbingly complex theorems and analyses which somebody took the trouble to prove. Take, for example, the formula for the area of a circle, pi*r2. This formula cannot be derived without a solid understanding of integrals. And guess what? Integrals involve limits, which are admissible only if you have infinite sets. They also involve heavy-duty trigonometry, too, without which one would not be able to simplify the equation in the integral to actually derive a concrete answer. (And anyone who has had to memorize trig identities will know just how "easy" they are...)

Not to mention that pi itself is an extremely non-trivial beast. Not only it is irrational; it is also transcendental, an extreme kind of irrationality which cannot be expressed by any algebraic formula. Such numbers cannot exist unless you have uncountably infinite sets (as opposed to, say, the set of natural numbers, which is infinite but still countable). Uncountable sets are extremely complex beasts with all kinds of bizarre properties. In fact, the Banach-Tarski decomposition arises partly from the uncountability of the set of points in ℜ3.

Whoopie. We started with the area of a circle, and bang we're looking at the Banach-Tarski paradox right in the face again.

So try as you may, you just can't get away, from mathemaaaatics! -- Tom Lehrer


[ Parent ]
area = pi*r^2 is not a good example (5.00 / 4) (#205)
by mindle on Sun May 25, 2003 at 06:17:43 AM EST

pi*r2 was derived by the greeks without a solid understanding of integrals and pi itself was estimated and used long before that by people who had no idead that it would someday be called trancendental.

[ Parent ]
Transcendental numbers. (5.00 / 3) (#207)
by i on Sun May 25, 2003 at 06:59:27 AM EST

Such numbers cannot exist unless you have uncountably infinite sets

This is not quite right. The set of computable numbers is countable.

and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]

Computable numbers (5.00 / 1) (#218)
by The Writer on Sun May 25, 2003 at 08:49:56 AM EST

Hmm, subtle difference here. The set of transcendentals is uncountable, IIRC, and pi is a transcendental. But I stand corrected that just because we cannot compute all transcendentals doesn't mean that we cannot compute some, like pi. OTOH, what exactly is meant by "computable"? Even numbers like e or pi aren't really "computable", in the sense that although we can approach them as a limit via an infinite series, can we actually compute them directly?

This does get into computability theory, which is rather obscure but very interesting. Maybe I should write my next story about the Busy Beaver numbers... (except the trolls might get a totally wrong idea)

[ Parent ]

Computable (5.00 / 2) (#223)
by i on Sun May 25, 2003 at 10:35:27 AM EST

means "computable to arbitrary precision". I would write "constructive numbers", but in constructive math the set of all numbers (which is the set of all computable numbers) is not countable.

How come? Constructivists insist that "set S is countable" means "there's a computable one-to-one correspondence between elements of S and natural numbers". Finding such for the set of all (computable) numbers is equivalent to solving the halting problem :(

and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]

Thanks (none / 0) (#239)
by The Writer on Sun May 25, 2003 at 02:25:26 PM EST

Now I'd be curious to know what numbers cannot be computed to arbitrary precision.

Now about constructivist proof of correspondence with the set of natural numbers... looks like they've added a nail in their own coffin. :-P Although it is interesting to find out that non-constructivity is related to computability theory in such deep ways.

[ Parent ]

Example. (5.00 / 3) (#254)
by i on Sun May 25, 2003 at 03:21:41 PM EST

Chaitin's constant (unsurprisingly :)

and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]
Chatain's Constant (5.00 / 2) (#263)
by pmc on Sun May 25, 2003 at 05:04:55 PM EST

Chatain's constant not only cannot be computed to an arbitary precision, it cannot be computed to any precision.

[ Parent ]
That should be "Chaitin's Constant" (none / 0) (#306)
by pmc on Mon May 26, 2003 at 08:27:50 AM EST

Sorry - finger/brain problem.

[ Parent ]
And another one (possibly) (5.00 / 1) (#311)
by pmc on Mon May 26, 2003 at 10:18:45 AM EST

Brun's Constant. This the sum of the recipricals of twin primes, and can be proved to converge, but nobody is quite sure to what. It's value can be calculated if the twin prime conjecture is true (and is equal to 1.902160583104....). To get the first decimal place by summing twin primes you need to add all the twin primes up to at least 10^530 - which is slightly beyond our capabilities at the moment.

So, until the twin prime conjecture is proved, then we only think this constant is about 1.9 (we know it is more than 1.82 though).

A bit of trivia - it was the calculation of twin primes to calculate this constant by Nicely that uncovered the pentium bug.

[ Parent ]

kick ass comment dude. (none / 0) (#394)
by ThreadSafe on Sat May 31, 2003 at 11:42:36 PM EST

Nice insight as well. This is why calculating pi to the nth using an infinite serious is crap. We all no what happens to diametricaly opposed orthogonal constructs as they approach an indefinate point!

Make a clone of me. And fucking listen to it! - Faik
[ Parent ]

kick ass comment dude. (none / 0) (#395)
by ThreadSafe on Sat May 31, 2003 at 11:44:05 PM EST

Nice insight as well. This is why calculating pi to the nth using an infinite series is crap. We all know what happens to diametricaly opposed orthogonal constructs as they approach an indefinate point!

Make a clone of me. And fucking listen to it! - Faik
[ Parent ]

I'll bet a dollar that if you cut open. . . (4.86 / 15) (#29)
by Pop Top on Fri May 23, 2003 at 03:34:58 PM EST

. . . a Banach-Tarski ball, you find some Greek geek dude named Zeno staring at ya' real purty like.

Just don't expect him to come meet you halfway!

+1 FP already (3.33 / 6) (#31)
by tetsuwan on Fri May 23, 2003 at 03:37:50 PM EST

People who cannot grasp the concept of infinity should go find another site.

Njal's Saga: Just like Romeo & Juliet without the romance

I cannot grasp the concept of infinity (nt) (2.78 / 14) (#37)
by circletimessquare on Fri May 23, 2003 at 04:07:41 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
I cannot grasp the concept of infinity (nt) (2.00 / 12) (#38)
by circletimessquare on Fri May 23, 2003 at 04:07:51 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
I cannot grasp the concept of infinity (nt) (2.00 / 12) (#39)
by circletimessquare on Fri May 23, 2003 at 04:08:01 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
I cannot grasp the concept of infinity (nt) (1.72 / 11) (#40)
by circletimessquare on Fri May 23, 2003 at 04:08:10 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
I cannot grasp the concept of infinity (nt) (1.63 / 11) (#41)
by circletimessquare on Fri May 23, 2003 at 04:08:19 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
I cannot grasp the concept of infinity (nt) (1.54 / 11) (#45)
by circletimessquare on Fri May 23, 2003 at 04:21:17 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
I can't either. . . n/t (3.66 / 3) (#60)
by Lew Dobbs on Fri May 23, 2003 at 05:29:08 PM EST



[ Parent ]
iS AnyBODY in HeRe *echo*echo*echo* <nt> (2.66 / 3) (#65)
by Dr Caleb on Fri May 23, 2003 at 05:49:25 PM EST


Vive Le Canada - For Canadians who give a shit about their country.

There is no K5 cabal.
[ Parent ]

Oh my God! (4.50 / 8) (#69)
by Kalani on Fri May 23, 2003 at 06:04:24 PM EST

I can see aleph-null from here!

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
MY GOD (3.00 / 1) (#156)
by Meatbomb on Sat May 24, 2003 at 01:49:54 PM EST

It's full of stars...

_______________

Good News for Liberal Democracy!

[ Parent ]
re: Oh my God! (5.00 / 1) (#204)
by ZorbaTHut on Sun May 25, 2003 at 06:12:03 AM EST

Let's sing a song to pass the time until we get there!

Aleph-null bottles of beer on the wall, aleph-null bottles of beer
Take one down, pass it around
Aleph-null bottles of beer on the wall!

[ Parent ]

to continue, (4.00 / 4) (#92)
by pb on Fri May 23, 2003 at 09:54:47 PM EST

click here.
---
"See what the drooling, ravening, flesh-eating hordes^W^W^W^WKuro5hin.org readers have to say."
-- pwhysall
[ Parent ]
Recursion is not infinity. (4.00 / 1) (#162)
by abulafia on Sat May 24, 2003 at 02:56:49 PM EST

Recursion may result in infinite behaviour, but they are not equivalent. At some point, this is nothing but semantics. Semantics are entirely meaninful. It reminds me of a recent dicsussion I had with someone about what storing a boolean without a NOT NULL clause in a database "means". I was pointing out that it has 3 possible values. I believe this is a provable fact. This seems to drive some people absolutely insane. I'm not trying to convert this conversation into an argument for constructivism. I'm only trying to point out that ontological differences are probably the root problem.

[ Parent ]
actually (1.60 / 5) (#184)
by circletimessquare on Sat May 24, 2003 at 08:45:32 PM EST

I'm not trying to convert this conversation into an argument for constructivism. I'm only trying to point out that ontological differences are probably the root problem.

actually, you have only proven you are a total dork ;-P


The tigers of wrath are wiser than the horses of instruction.

[ Parent ]

Feel free to bite me then. (none / 0) (#389)
by abulafia on Sat May 31, 2003 at 03:22:36 PM EST

I mean that in the kindest, most conversational manner. You can clamp your teeth around a limb, derive whatever fun you can from that, and enjoy the results.

I'll omit discussing various methods of folding chunks of your discussion and where they might go.

If you're interested in _responding_, that's different. I left plenty of seeds for conversation.

[ Parent ]

point taken (none / 0) (#377)
by adiffer on Thu May 29, 2003 at 02:37:04 AM EST

The environment in which I program at work uses three-valued booleans.  I have to change my way of thinking a bit, but it isn't hard once you get used to it.
--BE The Alien!
[ Parent ]
Hey, sir! (3.57 / 7) (#47)
by Random Number Generator Troll on Fri May 23, 2003 at 04:23:18 PM EST

Judging by the '2' rating you gave me, you took offense to my anti-infinity propaganda. I was merely commenting on the fact that infinity is not a number, has no real value, and hence any theorems constructed from it are inherently open to interpretation. This leads to theorems such as Starski and Banutch's.
However, it is clear that Circletimessquare does not understand the concept of infinity, as he has only posted ~6 replies, and should go and find another site.

[ Parent ]
Infinitely misunderstood (5.00 / 2) (#141)
by gidds on Sat May 24, 2003 at 10:43:57 AM EST

infinity is not a number, has no real value, and hence any theorems constructed from it are inherently open to interpretation.

I think this shows a basic misunderstanding of what mathematics is.  It is not the study of real-world objects.  It is the study of abstract patterns and forms.  Some branches may of course have powerful applications to the real world, but that's not why or how they're studied.

In mathematics, all that matters is whether a concept is well-defined.  If so, then that definition lets you use it precisely, and derive things from it, whether it concerns real, finite numbers, imaginary ones, infinite cardinalities, homotopy groups, n-manifolds, or whatever.

In everyday language, `infinity' is a very loose term; in mathematics, where infinite concepts are used, they are precisely defined, and so theorems constructed with them are just as valid as any other.  They may not correspond with objects in the `real world', but as I said, that's not the point.  In this case, the author was at pains to point out that the Banach-Tarski decomposition is not applicable to real-world, non-infinitely-divisible objects, only to idealised ones.  So I really don't understand your objection...

Andy/
[ Parent ]

Math isn't about numbers (5.00 / 1) (#159)
by abulafia on Sat May 24, 2003 at 02:44:35 PM EST

I'm not sure if you want to talk about K5 politics, or math. Assuming you want to talk about math, I'd suggest that you consider the fact that math isn't about numbers. It is about relations between various things, which sometimes look like numbers. That looks annoying and semantic on the face of it, but it actually is true. Set theory makes this a little clear by implication, without actually attempting to do so. An analogy to consider here is whether or not accounting is about money, or whether politics is about nation states. There is a layer of indirection that is important to consider.

[ Parent ]
bullshit (5.00 / 1) (#313)
by city light on Mon May 26, 2003 at 11:43:58 AM EST

Whenever mathematicians use the term infinity, they define exactly what they mean by it. For example the limit as x tends to inifinity of a series has a precise (but slightly messy) definition.

Also things like the Reimann Sphere contain a point labeled as 'infinity' (and yes it's just a convenient label) which is used to compact-ify the complex plane into something equivalent to a sphere. Sometimes when infinity is used it's given algebraic properties too. What matters is that in each context where it's used, there's never any confusion about what is meant by infinity.

[ Parent ]

Shit. I better take off. (1.00 / 1) (#211)
by Estanislao Martínez on Sun May 25, 2003 at 07:22:40 AM EST

Nevermind that the biggest minds in mathematics have struggled to grasp this concept, and many still do. Bye.

--em
[ Parent ]

Please name one that still does [nt] (5.00 / 2) (#226)
by i on Sun May 25, 2003 at 11:16:43 AM EST



and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]
there is no spoon (nt) (2.20 / 5) (#43)
by circletimessquare on Fri May 23, 2003 at 04:14:24 PM EST



The tigers of wrath are wiser than the horses of instruction.

It is frightening how much you want me. (1.00 / 6) (#46)
by Hide The Hamster on Fri May 23, 2003 at 04:22:00 PM EST




Free spirits are a liability.

August 8, 2004: "it certainly is" and I had engaged in a homosexual tryst.

[ Parent ]
you're trollicious (nt) (1.00 / 4) (#48)
by circletimessquare on Fri May 23, 2003 at 04:28:57 PM EST



The tigers of wrath are wiser than the horses of instruction.

[ Parent ]
Made me laugh,thanks :) <nt> (5.00 / 1) (#81)
by morceguinho on Fri May 23, 2003 at 07:54:56 PM EST



[ Parent ]
mathemeticians are inhuman (4.75 / 4) (#52)
by coderlemming on Fri May 23, 2003 at 04:43:00 PM EST

(in human language, they do not have a well-defined volume; it is impossible to measure their volume)

AHA! I knew it! Mathemeticians are inhuman!


--
Go be impersonally used as an organic semen collector!  (porkchop_d_clown)
Great article (5.00 / 4) (#72)
by marx on Fri May 23, 2003 at 07:02:23 PM EST

Even though I'm not really a mathematician, I have ended up working in a group of applied mathematicians, and this question is a bit relevant for me.

The senior members of the group (and almost everyone else) are constructivists. For a constructivist, something must be computable before he accepts that it exists. I think this is a very healthy attitude, and for someone who spends a lot of time computer programming it's quite natural. I.e. if you cannot even construct an algorithm for computing the object you want to introduce, why should anyone accept it, or want to discuss it further?

My interest lies in simulation of solid mechanics, and in that field appears what I think is a variant of this problem. The traditional models of solid mechanics (and mechanics in general) model bodies as a continuum. A body is precisely such an infinitely divisible object without a crystalline structure as presented in this article.

For real bodies, this crystalline structure can be interpreted as a definition of which atoms are significantly interacting and which are not. For a continuum you do not have this, and it leads to much seemingly unnecessary model complexity and workarounds.

My long-term aim is to do away with continuum mechanics, and replace it with discrete models, and I think this article provides me with some more good reasons for doing that.

Join me in the War on Torture: help eradicate torture from the world by holding torturers accountable.

Well.. (none / 0) (#88)
by Vann on Fri May 23, 2003 at 09:16:32 PM EST

What if a non-constructive proof (e.g., by contradiction) provides something can, in turn, be used constructively? Many times in mathematics you only need the existence of something to proove something else constructively.
____________
Sex is tedious all year except on Arbor Day. -- Rusty
[ Parent ]
then it doesn't count as constructivist (5.00 / 4) (#96)
by Delirium on Fri May 23, 2003 at 10:51:54 PM EST

The entire point of constructivism is that for every fact you claim as proven, you can construct an example. The original motivation for this was that as proofs were becoming incredibly complex, it was becoming difficult to trust them at all, since verifying every detail was nearly impossible (and in fact several proofs in major journals were later found to have subtle but very important errors in them). Constructivism solves this problem (at least partially), because verifying constructive proofs is significantly easier, as it's simply a matter of checking that the claimed construction does in fact satisfy the conditions it is claimed to satisfy.

[ Parent ]
Odd (4.66 / 3) (#111)
by Vann on Sat May 24, 2003 at 03:05:24 AM EST

That seems sort of silly. What about, say, the fundamental theorem of algebra? That's hardly constructive, since it tells you how many complex roots a polynomial has but doesn't tell you what those roots are. I'm sure you know that polynomials above a certain degree have no algebraic solution at all, so a "constructive" proof would "seem" to be impossible. Would a constructivist throw that out? Or am I just misconstruing what it means to be a constructivist?
____________
Sex is tedious all year except on Arbor Day. -- Rusty
[ Parent ]
Some Proofs of the FTA are constructive (4.83 / 6) (#136)
by Lupus Rufus on Sat May 24, 2003 at 09:32:54 AM EST

When I was learning about the FTA in school, what you stated as the FTA was considered to be an easy corollary of the main point, that a nonconstant polynomial F(x) with complex coefficients has a complex root r. From this you can conclude that (x-r) divides F(x) (the remainder of synthetic division is 0) and then by induction on the degree of F conclude that F has as many roots (with multiplicities) as its degree.

Now still, this seems on the surface to be a nonconstructive statement, I mean, how does one go about finding this mythological root of F? Well, that's what the proof is for! There are indeed proofs of this statement which can be interpreted as constructive, in the sense that they "zero in" on a root, forming a Cauchy sequence which converges to a complex number r with F(r) = 0. Since the reals (and hence the complex numbers) are constructed out of such Cauchy sequences, we can use this Cauchy sequence to construct an explicit complex number which is a root of F.

Actually, in general I think anything proved with the axioms of set theory without the axiom of choice can be interpreted in this way. The point being that one can wrap one's mind around all the existence statements in the axioms except the existence statement in the axiom of choice. It's an intuitive point, whether you want to accept that there are sets you cannot "understand" or not. But you certainly do not need these inexplicable sets to prove the fundamental theorem of algebra!

We believe in nothing, Lebowski, nothing.
[ Parent ]

Verifying how? (none / 0) (#137)
by CodeWright on Sat May 24, 2003 at 09:38:52 AM EST

On the basis of an algorithm or an implementation of an algorithm....?

I hope you don't say implementation, because there are peculiarities of implementation in chips and mathematical modeling programs...

--
"Jumpin Jesus H. Christ riding a segway with a little fruity 1 pint bucket of Ben and Jerry's rainbow fairy-berry crunch in his hand." --[ Parent ]
usually by a mathematician (5.00 / 4) (#148)
by Delirium on Sat May 24, 2003 at 11:55:28 AM EST

Certainly the mathematician can use tools if they'd be helpful. The main point is that the verification is possible at all. To use the other example brought up in this thread, a non-constructive proof of the fundamental theorem of algebra would simply tell you how many roots a polynomial has -- this is difficult to verify for complex proofs, because it means essentially checking every single step of the proof (and as proofs get enormous and full of subtleties, the chance of missing an error increases). If, on the other hand, you had a constructive proof that told you "this is how many roots a polynomial has, and here are the roots", then it's much simpler to verify that those are indeed the roots.

This is an oversimplified way to look at it, but for many problems the essential difference is between proving "there must exist some x, but I don't know what it'd look like" vs. proving "there must exist some x and this is what that x must be".

[ Parent ]

don't follow them into constructivism (4.83 / 6) (#97)
by martingale on Fri May 23, 2003 at 11:01:27 PM EST

Constructivism is seductive, but ultimately limiting.

It makes you think in terms of little objects you can construct, which is nice for some things people have already worked out, but what if the simplest construction is more complex than you have the patience to work out fully?

I've found that when it comes to proving results, this can be counterproductive. Many times, an answer arises by looking at things in a different way, as they are, not as we want them to be. Insisting on constructibility is just an extra limitation which often impedes switching to a different way of looking at things.

My long-term aim is to do away with continuum mechanics, and replace it with discrete models, and I think this article provides me with some more good reasons for doing that.
This I do not understand. What do you gain from a discrete model? Will your results be qualitatively different? Your ultimate aim is approximating reality, after all. It makes no difference if your approximate answer is an exact answer to an approximate problem, or an approximate answer to an approximate problem. What do you gain by switching to a much more cumbersome tool?

[ Parent ]
Discrete models (4.75 / 4) (#99)
by marx on Fri May 23, 2003 at 11:24:29 PM EST

What do you gain by switching to a much more cumbersome tool?
The point is that the continuum models are cumbersome, while the discrete models are simple. Describing the motion of a set of point masses is simple, while describing the motion of a continuum is hard, and seemingly unnecessarily so, since a body in reality is not a continuum, it is discrete.

I'm not a purist, so I'm not going to waste time painfully constructing some model just to prove a philosophical point, but I think simplifying things is meaningful.

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[ Parent ]

not at all (5.00 / 5) (#108)
by martingale on Sat May 24, 2003 at 02:11:45 AM EST

The point is that the continuum models are cumbersome, while the discrete models are simple.
I would suggest quite the opposite. Discrete models are full of nasty boundary cases and special cases, and all you have to work with are recurrences. A lot of differential equations can be derived by starting with a complex discrete model and dropping terms to simplify. Besides, continuous models don't usually include periodicities which can appear in discrete systems.

Naturally, this all comes down to what each person's favourite mathematical toolbox contains, so I'm not going to claim I'm on higher ground than you ;0)

One thing that's in favour of continuous models is that they are a lot better understood than discrete equivalents. If you're going to perform error analysis, that's a big advantage.

Describing the motion of a set of point masses is simple, while describing the motion of a continuum is hard, and seemingly unnecessarily so, since a body in reality is not a continuum, it is discrete.
That, I think, is not such a great reason. If you're going to treat a couple of particles together, that's one thing. But there are simply no natural tools to work with large ensemble of particles individually. Analysis will force you to either a continuous approximation, or a statistical approach. The latter is also a kind of continuous approximation. Why not skip the useless discrete setup?

Anyway, this is just turning into a rant on my part. Stay pragmatic, and don't restrict your toolbox on a point of principle. Every approach has its problems.

[ Parent ]

Error estimation (5.00 / 1) (#130)
by marx on Sat May 24, 2003 at 08:59:18 AM EST

If you're going to perform error analysis, that's a big advantage.
Yes, this is the big problem. What I will start with is try to show equivalence between a discretized continuum model and an initially discrete model, and then use the parts of whichever model or viewpoint fits best for each situation. I will probably have to use error estimates derived from continuum models.

Perhaps one can argue that there are no mathematical differences, but the difference is in the mindset. What I would like to get away a bit from is the analytical rigor which I think is limiting the continuum models. I think the more respectless approach found in for example computer graphics and discrete models (also some continuum models for that matter) is more productive.

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[ Parent ]

remember the ghost stories (5.00 / 3) (#142)
by martingale on Sat May 24, 2003 at 10:44:25 AM EST

What I would like to get away a bit from is the analytical rigor which I think is limiting the continuum models. I think the more respectless approach found in for example computer graphics and discrete models (also some continuum models for that matter) is more productive.
You probably are well aware that computer simulation is plagued with incorrect results due to discretizations which blow up, decay or behave weirdly unbeknownst to the program author.

This is usually no problem in computer graphics, since even an incorrect result looks good, but if accuracy is crucial, things can get tricky. If you write your code portably, you can run it on several different architectures (pc, supercomputer, etc.) and several different implementations to verify that you're seeing the same results. Can't say this is less work than the old estimation, tough...

[ Parent ]

Same problem (5.00 / 1) (#155)
by marx on Sat May 24, 2003 at 01:14:52 PM EST

You probably are well aware that computer simulation is plagued with incorrect results due to discretizations which blow up, decay or behave weirdly unbeknownst to the program author.
Yes, but this is the same whether you use a continuum model or a discrete model. To solve the continuum model you have to discretize it anyway, and at the end the two models will end up being very similar computationally.

Also, I do not dispute that time is continuous, so the time discretization will be the same for both solutions. It's the space continuum that I claim causes undue headaches. Usually it's the time discretization which causes the worst kind of artifacts and "blowups", but that is the same for both, and it is a very manageable problem, at least for the kinds of models I am interested in.

The continuum model describes the problem as a partial differential equation (PDE) (derivatives of both time and space variables) whereas the discrete model describes the problem as an ordinary differential equation (ODE) (derivatives of time). When you discretize space for the continuum model (using the finite element method for example), you end up with an ODE, which is similar (or equivalent in some meaning) to the ODE for the discrete model.

Basically the issues are that if I dream up some discrete model in this way, a) does it model reality well enough, and b) can I find some error estimate to be able to refine/coarsen the model. If I can prove these two things, then the world is my oyster. a) can be done experimentally, it's even enough to prove that it is close to a continuum model (because that has already been proven experimentally), and b) can probably be borrowed from a continuum model for the time being, and perhaps proven independently later on if the need arises.

I'm just not seeing the necessity of people spending literally years on cramming the intricacies of continuum models if there are simpler alternatives. It seems that there has been a sort of religious cult that has believed that a body really is a continuum, and then in a sort of "imperfect" world we have to accept that there are atoms, as some kind of measurement error or whatever. Maybe some people make a "leap of faith" when they first encounter calculus to be able to accept the concept of the continuum, and that they never really return to Earth. I have always thought calculus to be a bit bizarre, but I think if you think of it in a numerics sort of sense, then it's easier to accept as something real. I think it's good to remain a skeptic though, which I think this Banach-Tarski paradox proves.

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[ Parent ]

time (5.00 / 1) (#183)
by pb on Sat May 24, 2003 at 08:29:24 PM EST

Time isn't necessarily continuous, but for all (most?) practical purposes, it might as well be. Some things are so fine-grained that it isn't worth trying to make an accurate discrete model of them unless it's absolutely necessary. Therefore, you're necessarily left with an inaccurate continuous model, or an inaccurate discrete model. Good thing they're only models...
---
"See what the drooling, ravening, flesh-eating hordes^W^W^W^WKuro5hin.org readers have to say."
-- pwhysall
[ Parent ]
Maybe you can help me with this (5.00 / 1) (#238)
by Kalani on Sun May 25, 2003 at 02:21:56 PM EST

Aren't the three things (matter, space, time) mutually qualitatively constructed? That is, it seems like discrete matter requires discrete space (otherwise some space is in an undefined state between the boundaries of the smallest discrete chunk of matter). Also, discrete matter and discrete space seem to require discrete time (otherwise a chunk of matter travelling between two chunks of space is in an undefined state for fractional slices of time). So if that's right (they're all three mutually discrete or continuous), then a paradox like this would suggest that matter must be discrete, and that space and time must also be discrete. What's wrong with this?

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
discreteness in QM (none / 0) (#276)
by martingale on Sun May 25, 2003 at 10:16:25 PM EST

AFAIK, discreteness in QM is just a consequence of the discreteness of the spectrum of the operators used to measure quantities. There's no reason to think of matter/space/time as discrete like lego components, to be put together in preferred ways (is that what you're saying? I'm not sure I understand what you mean otherwise).

So, as long as you can devise measurement operators with discrete spectra, you're fine. But what about combinations of operators which give nondiscrete spectra? Does that mean they are not physical, or that the Quantum world isn't fully discrete?

It's easy (but wrong) to generalise the properties of toy models, which are designed for ease of exposition to undergraduates and/or technical simplicity.

[ Parent ]

PS (none / 0) (#242)
by Kalani on Sun May 25, 2003 at 02:38:07 PM EST

On the subject of a discrete model of space and time ... I have always thought that the notion of discrete space and time gives us sensible explanations for otherwise strange things. Like, why should the speed of light be constant? If space and time are discrete, then the constancy of the speed of light falls out as a natural consequence. It's one chunk of matter travelling to one chunk of space within one chunk of time. What really gets me is this idea that Wolfram has (and others have had) ... that maybe we don't even need matter at all. Maybe this discrete lattice of space is a free form network of spatial nodes ... and the matter that moves between nodes of the network is really closed networks of loops of space. Maybe chunks of "matter" are the rounding errors and nasty effects of discrete systems in the first place.

I know it's probably just idle dreaming by somebody not in the field, but there you go.

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
Discreteness (5.00 / 2) (#283)
by The Writer on Mon May 26, 2003 at 12:21:44 AM EST

Be careful how you interpret the quanta in quantum mechanics. While it's tempting to think of it as the "true atom" (in the original sense of the word "atom", which meant "indivisible"), packed into a well-defined region of space, this is unfortunately not a very accurate picture of subatomic particles as described in quantum mechanics.

Matter is a very strange beast at the subatomic level. Although it comes in discrete units, the discreteness is only at the energy level; it does NOT have a clear-cut spatial boundary. Rather, it comes as this bizarre particle-wave dual thingy, which peaks at a region we roughly understand as its "boundary". But instead of being strictly within that boundary like a lego block, it tapers off gradually, and arguably spreads throughout all of space-time, except that its effects quickly diminish into insignificance as distance from its "peak" increases, so for all intents and purposes, we treat it as though it were a lego block residing strictly within a certain spatial boundary.

Also, the motion of matter at the subatomic level is quite unlike our intuitive understanding of motion; an electron, for example, can interfere with itself and spread out into a waveform over a wide area, only to collapse at the last moment to a single point according to its probability distribution. That is to say, when a single electron passes through a double slit, it's as if it "randomly" changes its direction so that it ends up in different places each time, the aggregation of which forms a distribution consistent with how a wave would pass through a double-slit.

And then you have the weird action-at-a-distance effect of two particles that interacted in a way which is ambiguous according to the Heisenberg principle, and you can separate these particles by any arbitrary distance, and the act of observing one particle causes the other particle to "collapse" into a state which can only be decided by how you observed the first particle.

As for the discreteness of space-time, it is unfortunately also much more complicated than what might appear suggestive on the surface. One has to keep in mind that under General Relativity, the reference frame of a photon has zero time. That is to say, if you could ride on a beam of light at the speed of light, you would be simultaneously at every point in the path of the light. As far as you're concerned, 0 time passed to get from point A to point B, even if the actual distance from an outside observer's point of view is billions of light-years apart. In fact, if you were moving at the speed of light, spacetime collapses into a single point; all distances are zero as far as you're concerned. That is to say, the speed of light is not a simple matter of one unit of space per one unit of time; it is zero units of space per zero units of time in the reference frame of the photon, and yet it is a non-zero, finite constant in every other frame of reference.

[ Parent ]

Time (none / 0) (#269)
by marx on Sun May 25, 2003 at 06:12:43 PM EST

Well, I guess we don't really know if time is discrete or not. However, we do know that matter is discrete. People can talk about clouds of probability or whatever they want, but matter is still made up of separate entities of something.

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[ Parent ]

believing in models (5.00 / 1) (#277)
by martingale on Sun May 25, 2003 at 10:28:45 PM EST

Just a few points about your last paragraph. Much of what we know about continuous pde models is exhibited by the simplest examples of each family, elliptic, parabolic and hyperbolic. It's worth knowing those properties well, because they can be spotted in more complex pde operators.

Since continuous models are natural limits of the type of discrete models you prefer, they *do* have a real existence, as an approximation in the limit of better and better refinement of your mesh. so even if you started without knowing anything about continuous models, and decided to study discrete models exclusively, you would independently construct continuous models in the limit. This is a kind of post justification for studying the continuous pde models. They exist in the same way the reals exist, so to speak.

[ Parent ]

Occam (none / 0) (#284)
by marx on Mon May 26, 2003 at 12:36:52 AM EST

It's worth knowing those properties well, because they can be spotted in more complex pde operators.
Yes, much of the knowledge of PDEs is indeed useful. However, even here you have symptoms of the "continuum illness" if you will. During the relatively short time I have studied PDEs and solution methods, I have learned that there is no agreement on what a parabolic problem really is.

Some of the computation researchers have proposed a definition which is related to the growth of the numerical error (parabolic problems are simple to compute accurately, since they "forget" errors and sort of converge to one solution). But this approach is essentially looking at ODEs, at discretized problems. This enforces my skepticism of trying to accomplish something at the analytical or continuum level.

Since continuous models are natural limits of the type of discrete models you prefer, they *do* have a real existence, as an approximation in the limit of better and better refinement of your mesh.
This is very true, and I agree with this perspective. The continuum models should be used as tools for analysing the numerical part, and I will likely have to do so, but they should not be considered as the true physics models.

This may seem bizarre, but I think we should be honest and admit that most people are not geniuses, and even geniuses can easily get confused when there is too much complexity. If we can keep everything very simple, except possibly some part about error estimates or something similar, then the field will be less confusing, it will be open to more people, and progress will be faster.

Differential operators in several dimensions, tensors, etc. are not simple, and I think it's suprising that so few people seem to criticize the field for using these kinds of things, and so excessively. There seems to be a kind of culture in some parts of science that the one who presents the most difficult and convoluted equations has won, i.e. if you have succeeded in presenting something which no one else can fully understand, then you cannot be criticized. It should be the opposite; the one who can explain something in the simplest way should "win".

This is nothing new, it's simply Occam's Razor, but for some reason it seems to have fallen out of grace lately.

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[ Parent ]

you're right (5.00 / 1) (#286)
by martingale on Mon May 26, 2003 at 01:47:46 AM EST

Some of the computation researchers have proposed a definition which is related to the growth of the numerical error (parabolic problems are simple to compute accurately, since they "forget" errors and sort of converge to one solution). But this approach is essentially looking at ODEs, at discretized problems. This enforces my skepticism of trying to accomplish something at the analytical or continuum level.
Whatever works best, go for it. The fun thing about maths is that interesting results can come from the strangest quarters. Who would have thought that algebraic properties of pde operators are essential for well posedness? Darn Hoermander... (and Lie, of course, who started it)

There seems to be a kind of culture in some parts of science that the one who presents the most difficult and convoluted equations has won, i.e. if you have succeeded in presenting something which no one else can fully understand, then you cannot be criticized. It should be the opposite; the one who can explain something in the simplest way should "win".
Depends on the person. Some people are great communicators. Luckily, anyone can read published papers and point out flaws if they can find them. If the paper's unreadably obscure, it won't have an impact anyway, unless/until someone else rediscovers the same thing in terms that a critical mass of people can understand.

In many sciences, someone's reputation is often made by asking insightful questions in seminars anyway. It's not always necessary to be fully conversant in the language of the problem, either. Feynman is famous for asking simple, stupid questions which cut right to the heart of the issue. So unless there's really something important about the heavy mathematics, people who go for complexity as an end in itself don't stand the test of time. The first time they'll meet a really smart person, they'll be shot down.

[ Parent ]

You are not alone (5.00 / 3) (#144)
by Alex Buchanan on Sat May 24, 2003 at 11:01:16 AM EST

For real bodies, this crystalline structure can be interpreted as a definition of which atoms are significantly interacting and which are not. For a continuum you do not have this, and it leads to much seemingly unnecessary model complexity and workarounds.

My long-term aim is to do away with continuum mechanics, and replace it with discrete models, and I think this article provides me with some more good reasons for doing that.

This is not an uncommon feeling - I know I felt something similar when I first came across the Banach-Tarski Paradox. Clearly the axiom of choice must go!

Unfortunately, removing the axiom of choice causes even more problems than keeping it. And from a purely practical point of view, most of the mathematical technology that is used to describe quantum theory relies on results that use it.

A problem with your proposed project you will encounter is that important phenomena such as phase transitions don't appear in discrete models but do when one moves to continuum ones. (I will point out that my experience is with 2D quantum field theories, though I believe the situation is similar in higher dimensions).

So, in seeking simplification you have a lot of complications to deal with. :o)

[ Parent ]

Hmm, interesting (none / 0) (#168)
by Anonymous 7324 on Sat May 24, 2003 at 05:09:21 PM EST

A problem with your proposed project you will encounter is that important phenomena such as phase transitions don't appear in discrete models but do when one moves to continuum ones.

Any hints on how to find links that explain this in more detail? I'm pretty interested in phase transitions, but usually from a physical chemistry (either macroscopic or stat mech) POV, and I've paid fairly little attention to the maths for their own sake.

[ Parent ]

A sprinkling of urls (5.00 / 3) (#176)
by Alex Buchanan on Sat May 24, 2003 at 06:25:57 PM EST

Being slightly old-fashioned, I learned the little I know about phase transitions from attending research seminars and reading books. One book that sticks in my mind is "Introduction to Gauge Field Theory" by Bailin and Love, though you really need to have a handle on your maths or physics to cope with it.

Google is most helpful. The first link it gave me was to the introductory chapter of "Quantum Phase Transitions" by Subir Sachdev. It looks like a good place to get going if you're not put off bit a little bit of maths. It also introduces several commonly studied models which you could then do some searches for on the ArXive.org high energy physics (hep-th) pre-print server.

[ Parent ]

look up the Ising model (5.00 / 3) (#186)
by martingale on Sat May 24, 2003 at 10:16:06 PM EST

The Ising model is probably the simplest model exhibiting phase transitions, and there's a huge literature to go with it. Google for it.

[ Parent ]
A problem with this (5.00 / 2) (#246)
by Kalani on Sun May 25, 2003 at 02:50:20 PM EST

For a constructivist, something must be computable before he accepts that it exists.

I think that this requires you to get rid of the real numbers. I'm not a mathematician (or a physicist, or anything I guess) but you'll find that Alan Turing followed this thread immediately. He said that you could imagine any number (including reals) as a digit sequence, and that a computable number is one for which the nth digit can be computed. That's a great thing ... because it can satisfy Pythagorus. His problem with the reals was that they contained the irrationals ... and he had no intuitive way of understanding the irrationals. But see, Turing threw all of that out the window by introducing this notion of computable numbers (because hey, irrationals like pi and e are computable). So now you've got a computational framework for all theories of numbers right? Unfortunately no you don't, computable numbers are *still* a subset of the real numbers. Because, for example, you can define a real number that is the proportion of the integers that represent programs that will halt. That real number can't be computed (and there are many many more of them).

But I'm with you man. I say we just throw out the real numbers. Of course, non-mathematicians (non-anythings) like me can afford to be so barbaric as to say that because we haven't erected this vast intellectual framework that includes the reals (an intellectual framework that according to most is still not yet proved contradictory).

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
Math doesn't attract me... (4.00 / 1) (#82)
by morceguinho on Fri May 23, 2003 at 08:03:24 PM EST

...basically 'cos i need to do it to move on to 2nd year. Yet i liked your article.

Just a few questions:
First, we take E, and rename each member of E so that a number x is renamed to x divided by two. What do we get? We now find that E=N. E is by itself already infinite, since it's a subset of an infinite group, so there's no need to divide it by two, only trying to "find it's end" should prove infinity.
Also, the idea of an infinite ammount of subparticles of the math sphere confused me a bit: are they infinitely small or an infinite number? If there is an infinite number of little bits inside a sphere with radius 1, then you a mathematical equivalent of a blackhole since they "grow in number" inwards... well sorta... ok, i managd to confuse myself.

Proving infinity (4.66 / 3) (#128)
by The Writer on Sat May 24, 2003 at 08:50:46 AM EST

E is by itself already infinite, since it's a subset of an infinite group, so there's no need to divide it by two, only trying to "find it's end" should prove infinity.

It's not quite that simple, unfortunately. There are different degrees of infinity, mathematically speaking. It is not good enough to say that set X is infinite and set Y is infinite; we cannot conclude that X and Y are equally infinite just by that. We need to establish a 1-to-1 correspondence between the members of X and Y before we can confidently assert that X and Y have the same number of elements. This is what I implicitly did with the renaming trick.

And about infinite subparticles... mathematically speaking, there are uncountably many points within a sphere, and points are by definition dimension-less (or 0-dimension) entities which are infinitely small. It's sorta a generalization of real-world atoms, if you will. In a real-world ball, the smaller the atoms, the more atoms you need to make a ball of the same size; the mathematical sphere in one sense is just what happens when you reduce the size of atoms to 0: you will need an infinite number of atoms to keep the sphere at its original size.

[ Parent ]

Gotcha! (5.00 / 1) (#175)
by morceguinho on Sat May 24, 2003 at 06:23:49 PM EST

the smaller the atoms, the more atoms you need to make a ball of the same size; - or lessening the space between them... sorry i'm being picky! :) thanks for the explaination.

[ Parent ]
Infinitely Cool Math, +1 (4.00 / 2) (#86)
by randyk on Fri May 23, 2003 at 09:02:45 PM EST

And now I'm going to get drunk and think about inifinite sets.



ugh (3.13 / 22) (#90)
by turmeric on Fri May 23, 2003 at 09:26:17 PM EST

nice explanation but i mean... all it is is a word game. you basically say 'cut into finite pieces' in the beginning. well im sorry, an amorphous cloud of infinitely dense particles is not a 'finite piece' except to some jerk mensa brat who has decided to redefine 'piece' without telling you and then go 'ha ha yr stupid' like those montessori kids did to bart simpson.

thats like saying you broke the integers into a 'finite number of pieces' and put them back together to make two sets just as big. well, 'piece' tends to mean contiguous thingy. but if you pull out every odd number, that is not 'contiguous'. that is a shitload of little dots. you call that a 'piece'? who calls that a piece?

by then i was so annoyed i couldnt get through the 'axiom of choice'... which actually sounded interesting. .. . because it basically invovles math nerds fighting each other over something they arent sure is true or not. i just love it when science people realize they are no different than religion

so the terminology gets you, eh? (none / 0) (#112)
by jreilly on Sat May 24, 2003 at 03:49:25 AM EST

Well, why don't you reread the article, and replace every instance of the word "piece" with the word "set"? It works for the mathematically inclined, and for you!
I eagerly await your next objection.

Oooh, shiny...
[ Parent ]
no, it doesnt (none / 0) (#149)
by turmeric on Sat May 24, 2003 at 12:16:16 PM EST

the 'hook' is that your audience at first thinks how they could take scissors to a big sphere and rearrange it to get two spheres. this is a pathetic and lame attempt to make your article 'appeal to the masses'. sorry but the masses are not that dumb. condescension cannot be disguised.

[ Parent ]
umm... (none / 0) (#177)
by jreilly on Sat May 24, 2003 at 06:27:33 PM EST

My article?
Check the authors again, bub?
Sheesh, apparently the masses are that dumb.

Oooh, shiny...
[ Parent ]
haha (none / 0) (#210)
by Estanislao Martínez on Sun May 25, 2003 at 07:18:18 AM EST

thats like saying you broke the integers into a 'finite number of pieces' and put them back together to make two sets just as big. well, 'piece' tends to mean contiguous thingy. but if you pull out every odd number, that is not 'contiguous'. that is a shitload of little dots. you call that a 'piece'? who calls that a piece?

Heh. There is this very general phenomenon of "experts" hanging out with each other all the time (mathematicians, linguists, what have you), and over time developing very weird jargons based on completely commonplace words. Then they come back and tell those who are not in the trade that they've been wrong all along about whatever they use those words for.

Classic example (from Wittgenstein, IIRC): physicists around the 1920s "discovered" that chairs are not solid objects (since they are made out of atoms, which themselves are mostly empty space). Less classic example: linguists "discover" that words like "away" are not adverbs, but prepositions; hell, even more, the word "bush" is a preposition in Australia. More relevant example: mathematicians "discover" that the set of even natural numbers is "just as big" as that of natural numbers. The list is endless.

[...] it basically invovles math nerds fighting each other over something they arent sure is true or not. i just love it when science people realize they are no different than religion

Heh. You ought to study some mathematical logic. The biggest lesson of it, really, is that you can never justify it. Pretty much whatever axiom of classical logic you can give me, somebody has argued to abandon it for an alternative.

--em
[ Parent ]

The integers work even for your 'piece' (5.00 / 3) (#227)
by rvcx on Sun May 25, 2003 at 11:18:43 AM EST

While I am also a bit surprised that such a technical piece on pure math actually made the front page, I'm a bit disappointed at the number of k5ers who seem to want to blame mathematics for something they find confusing. I think tech posts can be equally obtuse to the uninitiated, but a reaction that computers are stupid and engineers are off their rockers is hardly mature.

If you have certain assumptions about what 'piece' means, then you've got to realize that you are the one with the extra restrictions. Frankly, I find the terminology 'set' similarly confusing--people usually don't think of a set of points as being infinite, while a piece of a continuous region does capture this possibility.

Further, while I'm not sure whether or not you can perform such a reconstruction with 'pieces' which meet your requirement that they be continuous, it might be possible, and you can certainly perform the 'integer' example with such continuous regions. You can consider both the positive and negative integers, and break the range into two pieces at zero. These original pieces are certainly continuous, although I freely admit that the reconstructed pieces (i/2 * -1 ^ i%2) don't look that way. However, if you're willing to think about the problem in two dimensions, you can just view a spiral starting at zero, then to one, then up and through the y axis (at a value of 1 or so) and down across the x axis at about -1, then down through the y axis near -2, and up through the x axis at 2. You've now taken a line which is only infinite in one direction (since the spiral has an endpoint at zero) and mapped it to the integer number line, which is infinite in two directions.

If you'd like to reject such notions as trickery, then that's your choice. I feel the author was very honest that this was an overview of an abstract mathematical notion and not a logic puzzle that anyone can be expected to solve without a grounding in advanced mathematics.

Sometimes I get the impression that 'Microsoft interview questions' have convinced many in the tech community that every hard question should be soluble with lateral thinking. That isn't the way the world works; sometimes you've got to crack a book.

[ Parent ]

speak english. this is america. (1.25 / 8) (#237)
by turmeric on Sun May 25, 2003 at 02:09:07 PM EST

just because something is in a different language doesnt mean you are smart. in fact, your complete inability to rewrite in language your audience can understand proves your stupidity. all great scientists understand the idea that communicating their work in human accessible words is important, in fact that is what they think science is, unraveling mysteries and bringing them down to the easily understandable level. if you were a super genius you might understand everything about the universe, but if you couldnt communicate it to anyone, nobody would care, nor should they.

[ Parent ]
Look man (1.06 / 29) (#93)
by BankofNigeria ATM on Fri May 23, 2003 at 10:01:24 PM EST

It's nice to see a pseudointellectual trying to fool people into thinking you're smart, but we see through your facade. -1, stop listening to emo and wearing berets.

FOR A GOOD TIME, AIM ME AT: Nigerian ATM

I care because? (1.00 / 12) (#101)
by Ta bu shi da yu on Sat May 24, 2003 at 12:09:42 AM EST

Recently I have noticed that many stories have been posted that are just not relevant on K5!

Remember that if I'm not interested, then it's not important.

Yours humbly,
Ta bù shì dà yú

---
AdTIה"the think tank that didn't".
ה

Back under the bridge! (3.50 / 6) (#102)
by Hide The Hamster on Sat May 24, 2003 at 12:22:23 AM EST

Troll!


Free spirits are a liability.

August 8, 2004: "it certainly is" and I had engaged in a homosexual tryst.

[ Parent ]
Offended by your "troll" implication (1.00 / 6) (#106)
by Ta bu shi da yu on Sat May 24, 2003 at 01:36:14 AM EST

Sorry kid, but there are those who are trolls and those who are not trolls.

Perhaps you should be targetting sdem? He is, after all, a much better candidate.

Yours humbly,
Ta bù shì dà yú

---
AdTIה"the think tank that didn't".
ה
[ Parent ]

Well, sport (1.66 / 3) (#146)
by Hide The Hamster on Sat May 24, 2003 at 11:33:42 AM EST

It seems that you just love to crawl my comment history!!! Maybe if you would have crawled a little deeper, you'd find this gem.


Free spirits are a liability.

August 8, 2004: "it certainly is" and I had engaged in a homosexual tryst.

[ Parent ]
Comment crawling (1.00 / 2) (#195)
by Ta bu shi da yu on Sun May 25, 2003 at 02:05:07 AM EST

I have much better things to do with my time than comment crawling. I appreciate you taking the time to point out that you have nothing to add to Kuro5hin.

Yours humbly,
Ta bù shì dà yú

---
AdTIה"the think tank that didn't".
ה
[ Parent ]

That violates physical laws! (4.00 / 1) (#105)
by tang gnat on Sat May 24, 2003 at 01:36:11 AM EST

But it doesn't matter (hehe) because nature is only approximated by R^3.

I thought.. (none / 0) (#120)
by Protagonist on Sat May 24, 2003 at 06:34:39 AM EST

..it was the other way around? :)

----
Hahah! Your ferris-wheel attack is as pathetic and ineffective as your system of government!
[ Parent ]
No, (none / 0) (#124)
by dcturner on Sat May 24, 2003 at 08:05:03 AM EST

There's no relation between the two. Ask a theoretical physicist.

Remove the opinion on spam to reply.


[ Parent ]
Congratulations on the FP! <nt> (1.00 / 4) (#109)
by morceguinho on Sat May 24, 2003 at 02:16:38 AM EST



God damn maths (2.40 / 5) (#110)
by TheOnlyCoolTim on Sat May 24, 2003 at 02:18:18 AM EST

It just sort of pisses me off that the same maths that describe the universe also give use weirdass shit like this. And it's even worse when the weirdass shit can describe the universe too, like imaginary numbers... Ninety degrees from reality but I understand they are used for several mathematical operations that we (indirectly) make use of every day...

Tim
"We are trapped in the belly of this horrible machine, and the machine is bleeding to death."

Imaginary numbers are used in everything. (5.00 / 4) (#114)
by subversion on Sat May 24, 2003 at 04:28:57 AM EST

They're inherent in most mathematical transforms; of course, the most obvious example would be the Fourier transform.  Simpler implementations throw away the imaginary content, but retaining it allows for some other tricks down the road.

Electrical engineers use the concept daily, as do people working in fluid dynamics, mathematicians, and probably many more people.

They're one of those concepts that seem counter-intuitive at first, but once you get used to/start using them they make a lot of sense.

If you disagree, reply, don't moderate.
[ Parent ]

Weirdass ! (5.00 / 1) (#115)
by jefu on Sat May 24, 2003 at 04:49:19 AM EST

It just sort of pisses me off that the same maths that describe the universe also give use weirdass shit like this. And it's even worse when the weirdass shit can describe the universe too, like imaginary numbers

You're not entirely alone in that - since the late 1800's mathematics has been plagued by a bunch of oddnesses that sometimes give the mathematicians the same kinds of feelings. Most of these ended up being foundational issues - just how do we build mathematical systems? And are they meaningful? Most mathematicians will tell you that their favorite systems (at least) are meaningful.

For a while there (around the 1900 mark) things got very confused indeed. And while the Axiom of Choice was not the only thing people did not agree on, the history of the Axiom of Choice gives the basic flavor of the controversies. Its out of print and very hard to find (libraries and their deaccessioning, ya know) but there's a fascinating and very readable book on the topic "Zermelo's Axiom of Choice: Its Origins, Development, and Influence" by Gregory H. Moore. It gets very mathematical in places, but most of that can be skipped without losing the flavor of things.

To address your comment though, mathematics is really just a way to describe things when you take away the things. 10 apples and 10 pencils and 10 songs - take out the apples and the pencils and the songs and you have the notion of counting and the number 10. Velocity is what you get when you drive your car down the road and take out the road and the car. The practical, applied side of mathematics stops here - physicists, engineers, and the like mostly just use the stuff that works and patch it up when necessary.

Mathematicians, though, while concerned about the practical aspects of things are also often concerned with things like "is this mathematical process really correct?" (Whatever correct means.) This is a reflection of the process in which mathematicians are repeating the abstraction process at a higher level, that is, taking the things (now mathematical entities) out of mathematics. Its often in this process that the fun stuff happens, where we find the odd foundational questions that don't matter (today at least) to the people who want to use the mathematics. But they do matter to the mathematicians who are trying to make sure that things really do work. (Parenthetically, one branch of mathematics has succeeded in taking most everything out leaving what is sometimes called "generalized abstract nonsense" or Category Theory - which now seems to be popping up in some odd places in theoretical physics. )

And just for fun and for those non-math-types who've struggled this far, the Axiom of Choice has several alternate and equivalent formulations (as well as a couple of non-equivalent variations) one of which is called Zorn's Lemma. Hollis Frampton, an experimental filmmaker in the sixties, made a film called "Zorn's Lemma". Its not likely to do well against "The Matrix - The Journey Home" (or whatever its called), but if you don't need much in the way of plot or characters you might find it rather wonderful.

[ Parent ]

it goes back even earlier (5.00 / 2) (#119)
by martingale on Sat May 24, 2003 at 06:13:19 AM EST

Disagreements about the fundamental objects go back further than the late 1800's. In the mid 1700's, mathematicians already discussed the meaning of functions, graphs, etc. Lots of theorems got proved and disproved and reproved because people disagreed whether a function was something you could/must draw with a pencil. The works of Euler and the Bernoulli family are full of such controversies.

Nowadays, a lot of definitions have settled, so it's not so noticeable.

[ Parent ]

Complex numbers are not really that strange at all (5.00 / 1) (#116)
by cgibbard on Sat May 24, 2003 at 05:06:22 AM EST

There's nothing really very strange about complex numbers at all. C is just R^2 (that is Cartesian product of the reals with themselves, the plane) together with a definition for addition and multiplication, that is a pair of functions (+):C^2->C and (*):C^2->C that satisfy certain properties.

Very nice theorems come out of the complexes and a lot of things about the reals (esp. in Analysis) are best understood by looking at the reals as a subset of the complexes.

Many people (I suspect most nonmathematicians) are simply unfamiliar with any sort of "number" other than subsets of the reals or complexes. In actuality, there are many other kinds of things that one might refer to as "numbers" to varying extents. Look up the mathematical definition of the terms 'ring' and 'field' if you're interested. There are much "stranger" structures available than the Complexes, for sure - structures where multiplication doesn't commute (you don't have to look far for these, acutally - matrices give an easy example), or things like non-Archimedean fields, where there exist numbers x for which 1/n > x for all naturals n (a bit trickier).

The fact that such things can be constructed doesn't detract at all from mathematics' beauty or usefulness. It is not a requirement that the objects constructed in a mathematical setting somehow model the real world, or even be visualisable. Just that the ideas surrounding them are interesting.

[ Parent ]

Complex numbers (none / 0) (#167)
by sigwinch on Sat May 24, 2003 at 04:45:22 PM EST

Indeed. Many interesting physical processes are best described using vectors in a Cartesian plane. Complex number notation is used purely for convenience: 4+5i is easier to write than 4X+5Y. (Especially when doing pencil-and-paper work, where boldface isn't available and you have to draw arrows on the top for X and Y. Ditto for computer programming.)

--
I don't want the world, I just want your half.
[ Parent ]

Math doesn't describe the universe. (none / 0) (#117)
by tkatchev on Sat May 24, 2003 at 05:33:34 AM EST

Just like a hammer doesn't describe how to build a house.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

But (none / 0) (#161)
by skim123 on Sat May 24, 2003 at 02:51:09 PM EST

It would be interesting if it did. And then one day some mathematician realized he had forgotten to carry the one, and, just like that, the universe's fundamental laws suddenly change so that the math (or at least our understanding of it), are reflected by the change.

Write a story about this, localroger, and I will read it. :-)

Money is in some respects like fire; it is a very excellent servant but a terrible master.
PT Barnum


[ Parent ]
err.. math *does* describe the universe (none / 0) (#165)
by neuropunk on Sat May 24, 2003 at 03:57:09 PM EST

Anything can be modeled with differential equations.
quod erat demonstrandum
[ Parent ]
Nah. (5.00 / 1) (#180)
by subversion on Sat May 24, 2003 at 07:01:16 PM EST

Math doesn't describe tkatchev's universe.  

Well, actually, yes it does.  We have irrational numbers, after all.

If you disagree, reply, don't moderate.
[ Parent ]

Anything? (none / 0) (#201)
by tkatchev on Sun May 25, 2003 at 05:09:53 AM EST

Anything in Newtonian solid-body mechanics, you mean.

Ghod I despise science-wannabees who don't know science...

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

I hate math wannabees (4.00 / 1) (#221)
by neuropunk on Sun May 25, 2003 at 10:04:17 AM EST

What, have you never heard of the Heaviside/delta-dirac function?
quod erat demonstrandum
[ Parent ]
No. (1.00 / 1) (#232)
by tkatchev on Sun May 25, 2003 at 12:47:49 PM EST

And I don't think I missed anything important.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

ignorant fool (none / 0) (#209)
by lester on Sun May 25, 2003 at 07:04:38 AM EST

then math does not describe the universe, we do. the world is contingent and mathematics aims to (pretends to?) discover necessary truths. you are confusing the realms of the mathematical and the empirical

[ Parent ]
I already answered this before I got this far comm (none / 0) (#393)
by ThreadSafe on Sat May 31, 2003 at 11:33:50 PM EST

http://www.kuro5hin.org/comments/2003/5/23/134430/275?pid=366#391

Make a clone of me. And fucking listen to it! - Faik
[ Parent ]

Cookie. (none / 0) (#118)
by tkatchev on Sat May 24, 2003 at 05:35:36 AM EST

Start here.

   -- Signed, Lev Andropoff, cosmonaut.

And ... (none / 0) (#125)
by Simon Kinahan on Sat May 24, 2003 at 08:17:27 AM EST

... what has constructivism got to do with it ?

Simon

If you disagree, post, don't moderate
[ Parent ]
everything (none / 0) (#208)
by lester on Sun May 25, 2003 at 07:02:13 AM EST

constructive mathematics does not accept results such as banach-tarski, which are nonconstructive. this because they appeal to the axiom of choice

[ Parent ]
I see what you mean (none / 0) (#231)
by Simon Kinahan on Sun May 25, 2003 at 12:19:53 PM EST

I'd always associated constructivism with rejection of proof by contradiction, but now you mention it I see why they'd reject the axiom of choice too.

Simon

If you disagree, post, don't moderate
[ Parent ]
not quite true (4.50 / 2) (#285)
by lester on Mon May 26, 2003 at 12:58:52 AM EST

this is a common misconception. constructivists actually do not reject proof by contradiction; they reject the double negation and excluded middle laws. a constructivist can perfectly well avail himself of proof by contradiction to prove -P, by assuming P and deriving a contradiction. what he can't do is assume -P, derive a contracition, and hold P to be proven, since this only proves --P. since he does not accept the double negation law, he can't conclude P from this. (excluded middle would also allow the nonconstructive step, since "P or -P" together with --P prove P). what constructivists reject is proving positive statements by contradiction; crucially, existential statements over infinite domains from their negation

the core of the matter, however, is the notion of infinity. classical math accepts actual infinities, constructive maths only potential ones. both kinds of math are equivalent over finite domains (since any statement about such a domain is trivially constructive). it is this which drives the technical differences in the logic-- the differences in axioms are chosen with this interest in mind

[ Parent ]

Another stupid question. (4.00 / 2) (#122)
by Verax on Sat May 24, 2003 at 07:10:32 AM EST

I've been struggling with this one for a while now.

Let's look at a 1D space: the real line. The set of positive reals has an infinite number of points. The set of the negative reals also has an infinite number of points. For every positive real number, there is a negative real number. So here's the question: isn't there some way to capture the notion that the set of all reals is twice as big as the set of the positive reals? Yes, I know they're both contain an infinite number of points. Yes, I know you can't, in general, compare infinities. But in this case, there's clearly a relationship where every point in the latter set can be associated with exactly two points in the former. Isn't there some way to express that there One set is twice as big as the other?



----------------------------------------------
"It is a poverty to decide that a child must die so that you may live as you wish." -- Mother Teresa of Calcutta
Measure Theory (5.00 / 3) (#123)
by dcturner on Sat May 24, 2003 at 08:02:54 AM EST

You can make statements like [0, 1] is half as big as [0, 2] in terms of measure (a generalised notion of the length of the set -- in this case you don't even need the generalisation) even though there is a bijection between them so they have the same cardinality. You could define some other measure on the reals so that the whole set had finite measure. For example, the standard measure of the principal value of arctan of your set gives the whole line a measure of pi, and the positive numbers have measure pi/2 as you want. But other sets won't behave so 'intuitively'.

Remove the opinion on spam to reply.


[ Parent ]
True (none / 0) (#295)
by liquidcrystal3 on Mon May 26, 2003 at 05:18:18 AM EST

Another way you could consider entire reals to be bigger than the positive reals would be by defining a measure which counted in how many ways the interval tended to infinity, hence [0,1] or any finite inteval would measure 0, the positives, 1, and the real line would have measure 2.

[ Parent ]
Comparing infinities (5.00 / 11) (#127)
by The Writer on Sat May 24, 2003 at 08:34:52 AM EST

Actually, there is a way to (rigorously and consistently) compare infinities, thanks to Cantor. The only caveat is that infinite quantities do not quite behave the way we might imagine from dealing with finite objects.

For one thing, just because you add more elements to an infinite set, doesn't necessarily mean that you will make it bigger. Similarly, just because you remove some elements from an infinite set, doesn't necessarily make it smaller (although it could).

For example, take the set of all integers N. If I remove, say, 50 elements from N, does that make it smaller? Not really: suppose we remove the first 50 numbers. Then I can simply go over the remaining numbers, and rename that to be (n-50), and I end up with N again. I can even remove an infinite number of elements from N, and it still remains the same size, as I illustrated from the odd/even example. However, if I removed all numbers greater than 1,000,000, for example, then the resulting set is no longer N; it is now a finite set.

Because infinite sets behave in this unusual way, we cannot generalize from the notion of "number of elements", because that really only applies to finite sets. Instead, we need to use a simpler definition of "size", due Cantor.

The basic idea is that we don't know how big an infinite set is; so we can't really assign a concrete "number" to it. However, suppose I could roll back time to when I was 3 years old and couldn't count up to 10 yet. Suppose I had a birthday party, where 10 people attended, and I wanted to make sure I had 10 hats to give each one of them. But since I couldn't count to 10, I couldn't count the number of people, and then count the number of hats, and check to see if they were the same number. What do I do? One easy way is this: simply give one hat to each person, and see if there are leftovers, or if there aren't enough. If there are leftovers, then I know the number of hats is greater than the number of people---I don't have to know what number it is! Similarly, if there weren't enough for everybody, then I know the number of hats is less than the number of people. I have just established a one-to-one correspondence between hats and people, and if it is possible to establish this correspondence that covers all hats and all people, then I can say with confidence there were as many hats as people, even though I couldn't count them.

Coming back to mathematical sets, this is how we compare (potentially infinite) sets. Given two sets X and Y, if I can establish a 1-to-1 relationship between the elements of X and Y, then I can say that they have the same size. If I cannot do this, then X and Y must be of different sizes.

For finite sets, this works as before, as we can verify by actually counting the elements. But for infinite sets, which strictly speaking we don't know the exact size of, we can still compare them using this method. For example, how many even numbers are there, is it half the number of integers? The surprising answer is that there are just as many even numbers as there are integers. This is easy to see: for every even number x, we associate it with (x/2). Similarly, for every integer y, we can make a unique even number by mapping y to y*2. This is, btw, exactly what I did when I did the "renaming" trick.

If we apply this to many infinite sets, we find that many infinite sets are equal in size. The number of integers is the same as the number of odds and the number of evens; similarly, the number of positive integers is the same as the number of negative integers (intuitive), but the negative integers themselves are just as many as the entire (positive and negative) set of integers. This may seem counter-intuitive, but remember that infinite quantities do not behave the same way as finite quantities. The very definition of infinite is that you can keep removing a finite number of elements from it, and it never runs out (if it did, it wouldn't be infinite).

By the same trick, we can also show that the number of rationals (fractions) is in fact, the same as the number of whole integers. I have a proof in one of my diaries, which I won't repeat here.

The question that might arise then, is whether all infinite sets are the same size. The answer is, in fact, no. It can be proven that it is impossible to have a 1-to-1 correspondence between the real numbers to the integers. There are infinitely more reals than there are integers. The proof of this is called Cantor's Diagonalization proof. I won't repeat it here unless you want me to, although it isn't very hard to understand.

So there are actually different degrees of infinity; and we should be careful when we use the word "infinity", since it is an imprecise word. It is better to give names to specific infinite sets, and say that other sets which have a 1-to-1 correspondence with it have the same size as the set. Conventionally, the size of the set of all natural numbers is called ℵ0 (pronounced aleph-zero or aleph-null). Any set that has a 1-to-1 correspondence with the natural numbers is said to have ℵ0 elements. The set of real numbers is much larger than ℵ0, however; we conventionally use c to denote the number of reals. We say that c > ℵ0 because while we can map each natural number to a unique real number (obviously), we cannot map every real into a unique natural number.

Coming back to your question, notice that the way we defined comparison of infinite magnitudes allows us to re-order elements in a set. When we proved that there are as many negative integers as there are integers, we re-ordered the negative integers so that they look like the set of all integers. Because this reordering is implicit in establishing the 1-to-1 mapping between sets, the set of negative reals are necessarily just as numerous as the entire set of reals. This way of counting is called cardinal counting, because we're dealing with unordered sets.

But there's another way of counting, called ordinal counting, where the order of elements in a set does matter. For example, if I take the number 0 from the set of integers, and stick it at the far end (i.e., after the rest of the numbers), then 0 becomes an "infinite" number, in the sense that you will never reach it by counting up from 1. We say that the order type of this set is ω+1 (ω is the order type of the set of natural numbers), and that it is one element "greater" (in the ordinal sense) than the set of integers. If we use this way of counting instead, then we find that the set of all (positive and negative) integers have a different order type from the set of positive integers. (OK, technically, the set of all integers doesn't have a well-defined order type, but the point is that when dealing with infinite sets, it matters whether or not you respect the ordering of elements.)

[ Parent ]

OT: Cultural bias. (none / 0) (#145)
by tkatchev on Sat May 24, 2003 at 11:15:06 AM EST

The stuff you just talked about in your post is basically what is taught where I live as introductory material to first-semester calculus students. (The cardinality of denumerable sets v.s. continuum sets.)

Is is different over at where you live?

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Math teaching methods (none / 0) (#151)
by abulafia on Sat May 24, 2003 at 12:34:33 PM EST

Cardinality and implications of continuum was something I learned in 11th grade, in German (exchange student) high school. Then I came back to a Tennessee school that spent an awful lot of time demonstrating how to do a proof in simple geometry. I find cardinality confuses a lot of people I work with when doing database development. These are folks with degrees from good schools. I don't think where you learned matters a lot, but whether or not you learned to think mathematically does.

[ Parent ]
Neat (none / 0) (#160)
by skim123 on Sat May 24, 2003 at 02:49:01 PM EST

This isn't something I learned until my junior year in college, taking a theory of computation class. (I hail from the US of A, btw.) Personally I've always found math classes, especially in high school, to just focus on memorizing equations and plugging and chugging away. Even at college, the math classes I took seemed to look more at application than theory and deeper understanding, but so it goes at a school known for engineering! :-)

Money is in some respects like fire; it is a very excellent servant but a terrible master.
PT Barnum


[ Parent ]
Memorizing equations (5.00 / 1) (#173)
by The Writer on Sat May 24, 2003 at 06:04:37 PM EST

One thing that really drew me to math during my high-school days is that my math teacher actually tried to explain things in a way that we understand why a certain formula is the way it is. I am no rote memorizer; in fact, I hate memorizing anything. Eventually, memorization has limited long-term utility, esp. in a such a complex area as mathematics. It is the understanding of why formulas are the way they are, why certain things are defined in that way and not another, that gives insights that can be useful for a long time to come.

[ Parent ]

Calculus (none / 0) (#164)
by jefu on Sat May 24, 2003 at 03:36:39 PM EST

I'm not really a mathematician - I studied math as an undergrad but I wandered over to CS afterwards. I'd probably have done OK in math and still have an interest in mathematical methods and mathematics in general.

So, I don't teach calculus (though I have) and I'm not sure what exactly is taught in calculus these days. But in a course where I was talking about the O(n) stuff in the analysis of algorithms, I said something like - "this kind of reasoning is almost exactly like that done in the epsilon-delta proofs in calculus". And quickly discovered that the epsilon-delta kind of proof seems to be disappearing from Calculus courses. Now, while the whole bit about counting things and countable vs uncountable sets always fascinated me, the epsilon-delta arguments seem to me to be the most important mathematical part of elementary calculus. And seriously elegant to boot.

Americans (at least) are seriously math phobic and are not encouraged to be otherwise by an educational system that seems to place more of a premium on keeping kids off the streets than actually teaching.

I learned Calculus partly from one of those general purpose calc courses, but mostly from the text "Calculus" by Michael Spivak which I would highly recommend to the motivated and talented student. Most students will just run away screaming. It not only does the usual derivative/integral stuff but also the epsilon-delta arguments, countable vs uncountable sets, a quick construction (via cuts) of the reals and even does a proof that e is transcendental. Do all the problems and you've covered a good bit of most undergraduate math curriculums. And, as is only proper, yellow pigs are considered.

[ Parent ]

association/mapping problems? (4.00 / 1) (#166)
by Verax on Sat May 24, 2003 at 04:14:03 PM EST

For finite sets, this works as before, as we can verify by actually counting the elements. But for infinite sets, which strictly speaking we don't know the exact size of, we can still compare them using this method. For example, how many even numbers are there, is it half the number of integers? The surprising answer is that there are just as many even numbers as there are integers. This is easy to see: for every even number x, we associate it with (x/2). Similarly, for every integer y, we can make a unique even number by mapping y to y*2. This is, btw, exactly what I did when I did the "renaming" trick.

Ok, I think this "associating" gets to the heart of my confusion. Doesn't this lead to a contradiction?

Consider someone adding balls into an urn. This someone mentally serializes each ball as it is added (every positive integer associates with a ball: the first ball is associated with 1, the second ball is associated with 2, and so on). So balls 1 through 10 are added, but then he removes ball 10. Then he adds balls 11 through 20, and then removes ball 20. This process repeats forever, so the number of balls placed into the urn becomes unbounded, and the number of balls taken out of the urn becomes unbounded.

Counting the number of balls in the urn after each iteration is easy. First there are 9, then 18, then 27, then 36, and so on. So let's consider the set of balls in the urn formed by letting this process go on forever. After each iteration, there are exactly 9 more balls than in the previous iteration. This set is not empty. Balls 1 through 9, 11 through 19, 21 through 29, and so on are never removed. This makes good sense to me.

However, here's the problem with associating numbers: Let's say someone else is doing the same thing. She also has an urn with unlimited capacity. She keeps pace with the original guy: for every iteration, she adds ten balls and removes one. However, she can't see his hand when he reaches into his urn, so she doesn't know which ball he is removing. So she does it a little differently. She serializes the balls the same way: the first ball added is associated with 1, the second ball added is associated with 2, and so on. However, she removes balls in the order that they were added. So she adds balls 1 through 10, and removes ball 1 for the first iteration. Then she adds balls 11 through 20 and removes ball 2 for the second iteration. Then she adds balls 21 through 30 and removes ball 3.

She does all this in lockstep with the first guy. The number of balls in the each urn are the same at each step. First there are 9, then 18, then 27, then 36, and so on. She considers the set of integers formed by the balls in her urn if the process were repeated forever. She reasons that for any positive integer n, it will have been removed. Since every positive integer maps to a ball that has been removed, she concludes that the set will be empty!

Something's clearly not right here. Observing how her set is constructed, the number of balls in her urn increases steadily with each iteration. If this goes on forever, the number of balls remaining in the urn is infinite. Yet she comes up with 0???. I can understand what the first guy does, and that seems correct. But her results don't make sense, even though I've seen her reasoning used in a math class before. Is it wrong? Does this show that mapping, used the way you did ("for every even number x, we associate it with (x/2)") is not correct? What am I missing?



----------------------------------------------
"It is a poverty to decide that a child must die so that you may live as you wish." -- Mother Teresa of Calcutta
[ Parent ]
The "repeat this infinite times" fallacy (5.00 / 3) (#171)
by The Writer on Sat May 24, 2003 at 05:58:58 PM EST

First of all, the "imagine this happens an infinite number of times" idea commonly used at high-school level math as an explanation of infinity is flawed. In mathematics, you cannot do something an infinite number of times, since the result of that is undefined. What is done is if you consider the completed set as a whole, and see what properties you can infer about this set. In other words, infinity is not "constructed" piecemeal---we can never reach it that way. Rather, we look at the infinite set as something that has already been completed, and see what behaviour we can infer from it.

That is why the common description of the set of natural numbers as the result of "adding the next largest number until forever" is misleading. The set of natural numbers is not the result of some ill-defined infinite operation on increasingly larger sets; it is the set that already contains all natural numbers by definition. Similarly, the set of real numbers is not the result of adding more and more numbers between numbers forever. (In fact, even if you could do this, you will not end up with the set of real numbers, only the set of rational numbers, which is only as large as the set of natural numbers.) It is the set defined to contain a certain class of objects, the real numbers. (Real numbers are rather complex beasts, I won't attempt to get into what they "are" here.)

The examples you describe do not describe infinite sets per se; but they are an example of something called the limit process. Intuitively, the limit process tries to find the boundary of a repeated operation if it were repeated indefinitely. Your first example has an infinite set as a limit; your second example has no well-defined limit. Note, however, that this doesn't mean the repeated process in the first example "creates" an infinite set; it doesn't, it merely has an infinite set as its limit. Similarly, the fact that the second example has no limit doesn't mean that somehow there's a contradiction; the existence of infinite sets does not require that they are limits of finite processes.

The right way to compare infinite sets is to treat them as already-completed entities, and to examine the correspondences between their elements.

[ Parent ]

Thank you. (none / 0) (#185)
by Verax on Sat May 24, 2003 at 09:53:34 PM EST

Thanks. It's clear that I've got some reading to do. But it's nice to know when something has been mis-represented to me, so that I know where to start working from. I can't claim that I completely "get it" now, but, thanks ot you, I have an ide of where to start looking.

As for "repeat this an infinite number of times", is there some place or reference that I can use to show that it's a fallacy? I'm certainly not well-versed enough in math to be able to show that on my own.



----------------------------------------------
"It is a poverty to decide that a child must die so that you may live as you wish." -- Mother Teresa of Calcutta
[ Parent ]
Debunking the "repeat forever" fallacy (5.00 / 3) (#190)
by The Writer on Sat May 24, 2003 at 11:03:16 PM EST

I don't know of any source that specifically debunks this fallacy. I guess you could approach it from the point that doing something "forever" isn't well-defined. What does "forever" mean? For example, suppose I have a machine that prints out 1 and 0 alternatingly. I let it run forever. What does it print after infinite time? Or, what does it mean to run the machine for infinite time? If infinite time is some point that can be reached, then the machine must print out either 1 or 0. But whether you say it's 1 or 0, I can always argue it should be the other instead. The problem is that infinite time can't be reached from finite time.

Another way of looking at this, is at the fact that what we usually think of as infinity (to be precise, ℵ0) is called a "limit" point (or a limit cardinal), which means you can never reach it from the finite side no matter how hard you try. In order to reach it, you sorta have to make a "quantum leap" from the finite to the infinite. This is what I was getting at when I said infinite sets are already completed; we cannot reach it from below, if we ever want to reach it, we have to "leap" to it directly.

In set theory, this "leap" is captured by the Axiom of Infinity, which states that there exists a set which contains all the natural numbers. The axiom doesn't define this set in the sense of "building" it from finite parts; it merely states that this set exists, contains 0, and if any number x is in this set, then x+1 is also in this set. (Notice it does not say to add x+1 to the set, but that x+1 is already in the set.)

The fact that an axiom dedicated to infinity is needed to reach what turns out to be the "smallest" infinity shows you how unreachable it really is.

[ Parent ]

Number of different infinites? (none / 0) (#352)
by JonesBoy on Tue May 27, 2003 at 11:29:56 AM EST

Great explanation!

I remember discussions about this sort of stuff in computational theory classes in college.   The prof said there were a certain number of infinite sets recognized, 7 if I remember right.   They were listed using hebrew letters.   I know it had something to do with the rate of expansion of expressions like x, x^n, x^x, et al.

Was this along the same track, or am I nuts?
Speeding never killed anyone. Stopping did.
[ Parent ]

Number of infinities (none / 0) (#363)
by The Writer on Tue May 27, 2003 at 01:49:12 PM EST

First of all, when dealing with infinite quantities in math, one must be careful to say whether one means ordinal infinities, or cardinal infinities.

Ordinals are a measure of sets where the order of elements are important. For infinite sets, this gives rise to the transfinite ordinals, such as ω, ω+1, ω2, etc..

Cardinals are a measure of the size of sets, without regard to the ordering of the elements. For example, the cardinality of the set of natural numbers is ℵ0; the cardinality of the real numbers is c. The aleph series that you refer to are related to sets of ordinals; so you have ℵ1, ℵ2, or even ℵ0. One of the very large cardinals in this series is called Θ, which is so large that ℵΘ = Θ.

As to how many infinities there are, the answer is, in fact, there are infinitely many. Given any cardinal X, we can take the set S which has cardinality X, and take all possible subsets of S (the powerset). The cardinality of the powerset always has a cardinality strictly larger than X. We denote the cardinality of the powerset of S by 2X, and we say that for all cardinals X, 2X > X.

In the same way, we can obtain increasingly larger ordinals using ordinal arithmetic; there is no upper bound to infinite quantities (pardon the pun).

[ Parent ]

Comparing Infinities (5.00 / 2) (#163)
by jefu on Sat May 24, 2003 at 03:15:58 PM EST

You have a way to generate two positive real numbers for every real (positive, zero, negative) number. Good. But that doesn't really do anything because there are ways to generate two real numbers (positive, zero, negative) for every positive number - which means that each of these sets is bigger than (or as big as) the other - so lets just say they're the same size.

Actually constructing such a function is left to the interest reader (meaning mostly that the easy examples would take mathematical notation which is not easy here, and the harder ones would take a lot of space).

But the real reason I'm writing this is to point out another of those fun facts having to do with the axiom of choice. You say :
you can't, in general, compare infinities (cardinals).
This is true if you do not use the Axiom of Choice, with the Axiom of Choice, you can always compare cardinals.

The Axiom of Choice can be such great fun.

[ Parent ]

short answer (5.00 / 1) (#244)
by city light on Sun May 25, 2003 at 02:41:23 PM EST

Depends what you mean by 'same size as'. There are various different mathemtical ideas which correspond roughly to this intuitive idea of the size of a mathematical object (cardinals, ordinals, measure, dimension, etc). Unfortunately (but interestingly) they don't always coincide. In terms of cardinality (which is all about one-to-one correspondance), the real line is the same size as the positive half of it. You can see this quite easily: Every point x in the real line has it's own unique 'partner' in the positive half of the real line, given by this function (and its inverse): y = e^x (x = log(y) the inverse)

[ Parent ]
Irrational numbers (4.00 / 1) (#134)
by kesuari on Sat May 24, 2003 at 09:30:25 AM EST

Pardon me for being off-topic, but it seems a few others are too :)

What is the point of irrational (and surreal and hyperreal) numbers? If I have a number x and I divide it by 2 (or 3 or 4 or 5 or ...), I've always got a number half as big as I started with. Why can't everything between 0 and 1 be rational?

Should be half (third, quarter, fifth ...) n/t (none / 0) (#135)
by kesuari on Sat May 24, 2003 at 09:32:01 AM EST

obviously.

[ Parent ]
Existence of irrational numbers (5.00 / 8) (#140)
by guffin on Sat May 24, 2003 at 10:22:36 AM EST

Lets prove sqrt(2) is irrational:

Assume that it is not irrational.  Then it can be expressed as p/q, where p and q are integers with no common factors (the fraction cannot be reduced).  It immediately follows that the square of the fraction, (p/q)^2 = p^2/q^2 also has no common factors.  

Now, we have the equation (p/q)^2 = 2.  Multiplying both sides by q^2 gives p^2 = 2 q^2.  But, this contradicts that they have no common factors!  Thus, our assumption is wrong, and irrational numbers exist.

[ Parent ]

Problem (none / 0) (#154)
by Eight Star on Sat May 24, 2003 at 12:52:39 PM EST

Couldn't the same process prove that sqrt(9) is irrational?

That makes all integers irrational


[ Parent ]

Nope. (5.00 / 3) (#157)
by czth on Sat May 24, 2003 at 02:31:13 PM EST

Taking the second paragraph (because the first would not change) from the proof that sqrt(2) is irrational:

Now, we have the equation (p/q)^2 = 2. Multiplying both sides by q^2 gives p^2 = 2 q^2. But, this contradicts that they have no common factors! Thus, our assumption is wrong, and irrational numbers exist.

For sqrt(9) this becomes:

Now, we have the equation (p/q)^2 = 9. Multiplying both sides by q^2 gives p^2 = 9 q^2.

However, the difference here is that for p^2 = 9 q^2, we can choose p = +/- 3 and q = +/- 1 to satisfy the conditions (that p and q have no common factors and that p^2 = 9 q^2), but we cannot do the same for 2. This mainly comes from 9 being a perfect square, e.g. p^2 = 9 q^2 = (3q)^2.

czth

[ Parent ]

No problem (5.00 / 6) (#158)
by skim123 on Sat May 24, 2003 at 02:44:15 PM EST

Couldn't the same process prove that sqrt(9) is irrational?

Nope. Try to use the previous poster's technique: We want to prove that sqrt(9) is irrational. Using a proof by contradiction, we assume sqrt(9) is rational, and we aim to find a contradiction (but, as you'll see, we won't). So, if sqrt(9) is rational, then sqrt(9) = a/b for some integers a,b. Squaring both sides, 9 = a2/b2, or 9b2 = a2, or (32)b2 = a2. Does this show a contradiction? No. After all, let a = 3 and b = 1 and everything works out...

In fact, you can use this technique to show that the square root of any non-perfect square is irrational. Watch.

Let n be a non-perfect square. We wish to show that its square root is irrational, and we'll do so via a proof by contradiction. Assume the negation of what we are trying to prove - assume that n is a non-perfect square and sqrt(n) is rational. Then, by the definition of rational numbers, sqrt(n) can be written as a/b, for some integers a, b. So, we have sqrt(n) = a/b, or, squaring both sides, n = a2/b2, or nb2 = a2.

Now, imagine that we express a and b in its prime factorization. Since we square both a and b, the number of exponents in each prime present in a and b's prime factorizations must be even. However, since n is not a perfect square, there is some prime factor of n that has an odd exponent. Therefore n * b2 will result in a number with some prime factor with an odd exponent. Therefore, if, indeed, a2 = n * b2, then these two supposedly equal numbers would have different prime factorizations. However, as we know from the unique prime factorization theorem, each integer's prime factorization is unique. Hence, a contradiction. Since we have found a contradiction, our assumption must be incorrect, meaning our initial claim, that if n is a non-perfect square it must be irrational, has been proven. QED.

Money is in some respects like fire; it is a very excellent servant but a terrible master.
PT Barnum


[ Parent ]
now I'm confused again! (3.25 / 4) (#215)
by dash2 on Sun May 25, 2003 at 08:26:20 AM EST

I appreciate that it is possible to find numbers for p and q. But the argument was that if the square root of 9 is rational, it can be expressed as p/q where p and q have no common factors (e.g. 3/1 for sqrt 9), and that therefore p^2 and q^2 also have no common factors. But

p^2 / q^2 = 9
ie p^2 = 9 q^2

ie they do have common factors!

I assume that the correct response is to say "ah, but q has to be greater than one, otherwise they don't really have common factors because one is not a factor."
------------------------
If I speak with the tongues of men and of angels, but have not love, I am become sounding brass, or a clanging cymbal.
[ Parent ]

Your assumption is correct (none / 0) (#323)
by skim123 on Mon May 26, 2003 at 02:20:29 PM EST

I assume that the correct response is to say "ah, but q has to be greater than one, otherwise they don't really have common factors because one is not a factor."

When talking about factors, mathematicians mean prime factors. Recall that every integer has a unique prime factorization. Since 1 is not a prime, 1 is not considered a factor in the prime factorization.

If 1 was considered a factor, then an algorithm for simplying a fraction to its simplest terms wouldn't really ever terminate. That is, say you had the fraction 4/8, and you want to simplify it.

  1. Ok, so you see both 4 and 8 have a factor of 2, so you divide both by 2 and get 2/4.
  2. Ok, both 2 and 4 have a factor of 2 so you divide and get 1/2.
  3. Now, both 1 and 2 have a factor of 1, so you divide by 1 to get 1/2.
  4. Go to step 3.

Infinite recursion! :-)

Money is in some respects like fire; it is a very excellent servant but a terrible master.
PT Barnum


[ Parent ]
I'll try it simple (3.00 / 2) (#304)
by Viliam Bur on Mon May 26, 2003 at 07:56:47 AM EST

Scenario for 2:

P^2 = 2 Q^2
This means P^2 is dividable by 2,
therefore P is dividable by 2 (P=2p),
(2p)^2 = 2 Q^2
4 p^2 = 2 Q^2
2 p^2 = Q^2
This means Q^2 is dividable by 2,
therefore Q is dividable by 2 (Q=2q),...

Stop! If both P and Q are dividable by 2, they are not the "smallest possible integers", because p/q are smaller.

Scenario for 9:

P^2 = 9 Q^2
This means P^2 is dividable by 9,
therefore P is dividable by 3 (P=2p),...

This is the difference! P^2 being multiple of 2 requires P being multiple of 2. P^2 being multiple of 9 requires only P being multiple of 3. So you get only P=3p, p^2=q^2, and that's the end.

[ Parent ]

What's the square root of -1? (none / 0) (#143)
by SlickMickTrick on Sat May 24, 2003 at 10:57:05 AM EST

It does not exist in the set of rational numbers. Which means we need more numbers to answer the question.



[ Parent ]
Doesn't exist in the rational numbers... (5.00 / 2) (#152)
by vectro on Sat May 24, 2003 at 12:50:19 PM EST

... but doesn't exist in the reals, either. This is a better answer to the question of "why can't all numbers be real?"

“The problem with that definition is just that it's bullshit.” -- localroger
[ Parent ]
Easy one (none / 0) (#147)
by mcgrew on Sat May 24, 2003 at 11:37:39 AM EST

So we can make pretty charts and graphs!

"The entire neocon movement is dedicated to revoking mcgrew's posting priviliges. This is why we went to war with Iraq." -LilDebbie
[ Parent ]

you need irrationals to do calculus (none / 0) (#241)
by city light on Sun May 25, 2003 at 02:31:40 PM EST

What's the point of them? From a practical point of view: trying to model real-life concepts and solve mechanical problems often involves solving differential equations - we need concepts like continuity and differentiability, to do this.

The field of rational numbers isn't sufficient to even define these concepts properly... (well, I think you can do a kind of analysis with just the rationals (?), but its a lot weaker than real analysis in terms of what you can prove from it, so isn't really much use). Read up about the completeness axiom for the real numbers - it's what gives the extra clout needed to prove everything we need to do calculus properly.

For example, functions like e^x, sin x, cos x, and (the obvious example) sqrt(x) don't exist as functions from the rationals to the rationals, but are nevertheless things we want to study and are very useful for solving problems

Or maybe this would help more: calculus is all about limits. We need to be sure that if a sequence gets closer and closer and closer to something (avoiding the technicalities), then that 'something' (the limit) actually exists in the set we're working with.

If we only had rational numbers, limits like this wouldn't exist. eg the limit to

3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...


[ Parent ]

problem of countability (none / 0) (#280)
by swagr on Sun May 25, 2003 at 11:41:07 PM EST

namely than rationals are countable while irrationals are not. Cantor's arguement is pretty easy to grasp (look it up somewhere), and mapping rationals to integers is pretty straight forward as well x/y --> x^3 * y^2. Once you do that, you see rationals and irrationals are very different sets.

[ Parent ]
Naive question (2.00 / 3) (#169)
by John Miles on Sat May 24, 2003 at 05:36:07 PM EST

I don't see the paradox. If you construct two spheres from the pieces of a single sphere, those two spheres will be at least partially hollow. So you're comparing the volume of a solid sphere with the volume of two semi-solid ones. (Which would be a ridiculously-elementary error, so I'm sure that's not what's going on.)

What am I missing?

For so long as men do as they are told, there will be war.

Uninformed reply (5.00 / 2) (#170)
by piter on Sat May 24, 2003 at 05:55:02 PM EST

Well, that's exactly right. Since the two resulting spheres are each made from one original sphere, then they will each be half as "dense," or constructed from half as many points as the original one. However, given that this is a mathematical sphere, it consists of all points at a distance from 0 to 1 from the origin. However, this is an infinite number of points, so the two spheres that we get at the end is half as dense as the original, infinitely dense sphere. So what's half of infinity? Infinity. Meaning that the two product spheres are each infinitely dense as well, and occupy the same volume as the original one, making them, in essence, two copies of the original.
"That that is is not not that that is not is not." Jacques Derrida
[ Parent ]
Mostly correct (none / 0) (#181)
by The Writer on Sat May 24, 2003 at 07:08:07 PM EST

The paradox, like you said, is that the two resulting spheres are identical to the original, i.e., they are completely solid, with no hollow areas introduced.

The whole reasoning about density is just my rationalization of it. Mathematically speaking, the two resulting spheres just happen to be exact copies of the original. Hence the paradox: how can you take effectively half of something, and reassemble it into the whole?

[ Parent ]

By cheating (2.00 / 1) (#193)
by rmn on Sun May 25, 2003 at 01:29:36 AM EST

Once you introduce the concept of infinity, you can do pretty much anything you want (x/n = x, x+n = x).

Likewise, you can say that if you start with zero spheres, you can easily double or triple that number (0*n = 0).

It's only a paradox if you think about them in physical terms. If the spheres were physical (and admitting you can get infinitely dense matter), then you could never get two with exactly the same properties as the original. They would nave to be partially hollow, or less dense, etc..

Now, the really important question (and the problem I believe Einstein was working on when he died) is: if you manage to halve human stupidity, will it actually make any difference...? ;-)

RMN
~~~

[ Parent ]

Correction (none / 0) (#194)
by rmn on Sun May 25, 2003 at 01:53:59 AM EST

Obviously in the message above I meant "admitting you can't get infinitely dense matter", not can.

RMN
~~~

[ Parent ]

So basically (none / 0) (#188)
by dipierro on Sat May 24, 2003 at 10:56:58 PM EST

it's the geometrical version of saying that there are the same number of even numbers as there are numbers?

[ Parent ]
re: so basically (none / 0) (#220)
by SlamMan on Sun May 25, 2003 at 09:35:14 AM EST

close enough. Its been a while since I've done any work with infinity theory, but mathmatically the infinity between 0 and 1 is different from between 0 and the end of rational numbers. They're both an infinite numbers of numbers, but ine more infinite.

[ Parent ]
yeah... (none / 0) (#222)
by dipierro on Sun May 25, 2003 at 10:12:12 AM EST

The number of integers is countably infinite, but the number of reals is uncountably infinite. So there are more reals than integers. But both the number of integers and the number of even integers (as well as the number of primes, for example) are countably infinite. There are the same number of even integers as there are integers.

The proof in that case is much simpler though. You have to make a one to one, onto mapping. I'm not going to get into the details.



[ Parent ]
The spheres aren't hollow (1.00 / 1) (#252)
by p3d0 on Sun May 25, 2003 at 03:11:58 PM EST

The two new spheres both include all points within their radius, just as the original did. Therefore, the two spheres are each identical to the original. That's the paradox.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
OK, but... (none / 0) (#267)
by John Miles on Sun May 25, 2003 at 05:47:22 PM EST

... you can do a LOT of things with mathematical abstractions that aren't meaningful in the real world, like taking the square root of a negative number, working with infinite and infinitesimal quantities, and so forth. What, exactly, makes this particular abstraction a 'paradox'?

For so long as men do as they are told, there will be war.
[ Parent ]
Check the dictionary (3.00 / 2) (#279)
by p3d0 on Sun May 25, 2003 at 10:39:03 PM EST

A paradox is a seemingly contradictory statement that may nonetheless be true. To think you could discard half the points in a sphere and still have the whole sphere seems contradictory; nevertheless, it is true. Ergo, it is a paradox.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
It's arguable... (none / 0) (#305)
by synaesthesia on Mon May 26, 2003 at 08:09:03 AM EST

...that a concept such as 'taking the square root of a negative number' is seemingly contradictory, if you start with the premise that 'the product of any two numbers which have the same sign is positive'.

Sausages or cheese?
[ Parent ]
Your point? (none / 0) (#330)
by p3d0 on Mon May 26, 2003 at 06:29:44 PM EST

So what? Then that's also a paradox for you.

Look, I didn't make up the word.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]

I'm not disagreeing with you. (none / 0) (#340)
by synaesthesia on Tue May 27, 2003 at 04:08:48 AM EST

Just pointing out that the line between 'interesting result' and 'paradox' is a little blurred.

Sausages or cheese?
[ Parent ]
Bah (3.33 / 6) (#172)
by Hoo00 on Sat May 24, 2003 at 06:03:35 PM EST

It basically says
inf = inf / 2 + inf / 2

An anology might be: if this article were infinitely difficult to understand, it won't make any sense to you even though you have skillfully divided it into two pieces, or study it twice or whatever.

don't dismiss what you don't understand (4.00 / 4) (#206)
by lester on Sun May 25, 2003 at 06:56:58 AM EST

It basically says inf = inf / 2 + inf / 2

except that the author has some clue what they are talking about, and you don't

a zen master comes across a boy who was trying to get a very stubborn mule to walk ahead by whipping it. the master said to the boy: "you can't get an animal to walk by whipping it. the only thing that will allow you to master the animal is understanding it." the master then grabbed the whip and whipped the mule, which immediately started walking.

[ Parent ]

Shall we always accept what we don't understand?nt (none / 0) (#362)
by Rezand on Tue May 27, 2003 at 01:43:30 PM EST



[ Parent ]
Shall we, instead, dismiss it? [nt] (5.00 / 2) (#368)
by Control Group on Tue May 27, 2003 at 04:38:23 PM EST



***
"Oh, nothing. It just looks like a simple Kung-Fu Swedish Rastafarian Helldemon."
[ Parent ]
of course, not (none / 0) (#376)
by lester on Wed May 28, 2003 at 04:21:51 PM EST

the only things worth dismissing are those we understand

[ Parent ]
Old joke (4.77 / 9) (#179)
by gjm11 on Sat May 24, 2003 at 06:46:52 PM EST

Q: I say, I say, I say, what's an anagram of "Banach-Tarski"?

A: "Banach-Tarski Banach-Tarski".

On a slightly more serious note: anyone interested in this stuff and with a bit of mathematical knowledge (say, first degree level) should read Stan Wagon's beautiful book called "The Banach-Tarski Paradox".



Equivalent to the axiom of choice (5.00 / 3) (#192)
by Repton on Sun May 25, 2003 at 12:40:32 AM EST

Q: I say, I say, I say. What's yellow and equivalent to the axiom of choice?

A1: Zorn's Lemon!

A2: The Banana-Tarski paradox!


--
Repton.
They say that only an experienced wizard can do the tengu shuffle..
[ Parent ]

Hmm (5.00 / 4) (#214)
by gjm11 on Sun May 25, 2003 at 07:59:41 AM EST

<pedant>Actually Banach-Tarski isn't equivalent to the axiom of choice; BT can work even if AC fails. I forget exactly how much Choice you need to get B-T, but all the details are in Wagon's book.</pedant>

A friend of mine once attempted to produce an exhaustive list of mathematical fruit jokes. There's the Zorn's Lemon one above, there's "What's purple and commutes?" (an abelian grape), and -- much more obscure -- there's "What's green and determined up to isomorphism by its first Chern class?" (a lime bundle). I don't think he found any others. So I think the Banana-Tarski joke is an original contribution to the field; you should fix up the minor error about how much choice it's equivalent to, write it up and submit it to J. Math. Fruct. Hum.



[ Parent ]
BT and AC (none / 0) (#217)
by The Writer on Sun May 25, 2003 at 08:37:48 AM EST

About the equivalence of Banach-Tarski and AC, I believe it depends on which system you're working with. I think under ZF without AC, BT is equivalent to AC. But I could be mistaken...

OTOH, paradoxical decomposition is not unique to systems with the Axiom of Choice; check out the PDF linked by i, it describes how you can obtain paradoxical sets without using AC.

[ Parent ]

Banach-Tarski and the axiom of choice (5.00 / 2) (#255)
by gjm11 on Sun May 25, 2003 at 03:33:32 PM EST

Yes, I think you're mistaken.

Banach-Tarski only requires Choice for a fairly small number of fairly small sets. (Specifically, for the orbits of the action of the free group on two generators on S^2. That's continuum-many countable sets.) I'm fairly sure "AC for <= k1 sets of size k2" doesn't imply full AC for any k1 and k2.</p>

[ Parent ]

Why are these named after people? (1.12 / 8) (#189)
by Fen on Sat May 24, 2003 at 10:58:45 PM EST

Duh, ideas exist outside of time. So Banach and Tarski thought of it after a particularly exausting bout of sex together. Did nobody in the entire world ever think of it before? What about other planets?
--Self.
Epistemology and scientific practice (4.50 / 2) (#202)
by xunam on Sun May 25, 2003 at 05:33:49 AM EST

Though this is kind of off topic, it deserves a comment. The question whether mathematics exist independently of time and human culture or not is one of the primary questions of epistemology. After all, most concepts in math were invented by people to formalize some intuition. The subject is wide open for debate, but this exceeds the scope of this comment.

Now about this paradox being named after Banach and Tarski. It is named after both because either they worked together on it or they discovered it at the same time (as it happens quite often), I don't know which it is in this particular case. Maybe someone thought of the paradox before, but they were surely the first to publish something about it, and that is what matters in mathematics. The reason why many important theorems are named after people is that it is clearer and easier to talk about the “Banach-Tarski paradox” that about the “paradox that says that a sphere can be cut into a finite number of pieces and recomposed into two spheres of the same size”...

-- The meta-Turing test qualifies an entity as intelligent if it applies Turing tests to systems of its own creation.



[ Parent ]
It's not wide open. (none / 0) (#271)
by Fen on Sun May 25, 2003 at 07:11:14 PM EST

The only reason people think someone "invented" some mathematical concept is because they can't get themselves away from their want to "invent" something new.
--Self.
[ Parent ]
not a maths student then (5.00 / 3) (#236)
by city light on Sun May 25, 2003 at 02:08:17 PM EST

Why? because if theorems (or apparent paradoxes for that matter) weren't named after people, they'd be called exciting and memorable things like 'Theorem 5.11 part (a)'. Which is a great memory aid when revising I assure you. So, yes, theorems exist outside of time, but humans and the process of mathematical discovery don't exist outside of time. We need some cultural and historical handles on the material we're studying, or else we'd be lost.

[ Parent ]
joke (5.00 / 2) (#240)
by The Writer on Sun May 25, 2003 at 02:27:59 PM EST

Or they could be named after their Godel number, which makes them even harder to remember. :-P

[ Parent ]

Godel number? (none / 0) (#266)
by Fen on Sun May 25, 2003 at 05:45:49 PM EST

Somethings are just hard to shake I guess.
--Self.
[ Parent ]
Axiom of Choice (5.00 / 5) (#196)
by gniv on Sun May 25, 2003 at 02:26:35 AM EST

This was a nice story. I learned something today.

However, I think your formulation of the axiom is not quite right (as pointed out by my wife):

If you're unfamiliar with the Axiom of Choice, it basically goes like this: if you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C.
(emphasis mine) If you take the sets {a,b}, {a,c}, {b,c}, there is no set that contains precisely one element from each of the three sets. I think you mixed two formulations of the axiom:

1. Given any set of mutually exclusive nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets (definition taken from here).

2. Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection ( definition taken from here).

The condition that exactly one element is taken from each set can only be imposed on mutually exclusive sets. The first formulation is due to Zermelo, while the second one predates him. What I'm not sure is whether the two are really equivalent.

Vague wording? (none / 0) (#216)
by The Writer on Sun May 25, 2003 at 08:30:37 AM EST

OK, perhaps I didn't phrase the definition correctly.

What I meant by "precisely one element from each set" is that you pick one, and only one, element from each of the sets. It is perfectly possible that the same element may be picked from two different sets, but that doesn't matter because duplication is ignored in sets. I guess your (2) is a better wording of this.

[ Parent ]

You just forgot... (5.00 / 2) (#224)
by zoopmf on Sun May 25, 2003 at 10:53:19 AM EST

to require that the sets be mutually exclusive.

gnev's example {{a, b}, {a, c}, {b, c}} shows clearly that H cannot always exist if the sets are not mutually exclusive.

[ Parent ]

Ah, I get it (none / 0) (#243)
by The Writer on Sun May 25, 2003 at 02:41:07 PM EST

Thanks for pointing that out.

[ Parent ]

Examine the other comments (5.00 / 1) (#347)
by wurp on Tue May 27, 2003 at 10:35:52 AM EST

Note that there is no set that has exactly one member from each of the sets listed in the parent comment.  So he's not just pointing out a lack of precision in your definition, but an error.
---
Buy my stuff
[ Parent ]
Uh.. (1.50 / 10) (#197)
by reflective recursion on Sun May 25, 2003 at 02:34:54 AM EST

The paradox states that it is possible to take a solid sphere
A solid sphere, eh? Okay..
rearrange them using only rotations and translations, and re-assemble them into two identical copies of the original sphere
Copies, eh? You say identical?
"Impossible!" I hear you say. "That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox.
So you are saying that mathematicians now realize that this process does not violate physical laws and they believe it is entirely possible to split a physical sphere into sections, rearrange those sections, and have two identical physical spheres?

-- Click --
First and foremost, we're talking about a mathematical sphere, not a physical sphere
Suuuuure. First and foremost, you don't have the slightest clue what you're talking about. Is confusion by metaphors your intention here? Or am I simply missing your outstanding brilliance?

re / (5.00 / 3) (#219)
by SlamMan on Sun May 25, 2003 at 09:28:12 AM EST

Nope, just missing the meaning of his words. Solid and Physical don't mean the same thing in his context. Solid means 'not just a shell.' Physical means 'could exist in real world.'

[ Parent ]
Not quite... (1.33 / 3) (#225)
by reflective recursion on Sun May 25, 2003 at 10:54:03 AM EST

"Impossible!" I hear you say. "That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox.
Ok, so you are saying that a solid sphere, that "could exist in the real world" can be split into sections, rearranged, and form two identical spheres... that "could exist in the real world." Nothing has changed in what I am saying, with your "new" definition of "physical."

The author did use a convincing bit of sophism to get readers. I'm just sad I fell for such propaganda myself.

[ Parent ]
The difference is this (5.00 / 3) (#230)
by Kalani on Sun May 25, 2003 at 11:32:55 AM EST

In the real world, spheres (solid or not) are composed of atoms and are fundamentally discrete. In the real world we work with finite collections of finite sets (so weird aspects of the axiom of choice aren't intuitive to us -- in fact we say that the axiom is just obvious when it's presented in its finite form). The reason that people (rightfully) reject the Banach-Tarski paradox on physical grounds is that it relies on certain assumptions that actually don't apply to the physical world. However, mathematicians don't generally care about the physical world (unless they're working under a system of axioms that are explicitly constructed to mirror the physical world in some way). So they imagine a world where matter actually *is* infinitely divisible and where the axiom of choice holds for infinite collections of infinite sets, and a consequence of that is this Banach-Tarski paradox.

Anyway, I'm not a mathematician so this point is probably obvious to everybody else.

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
Of course.. (1.33 / 3) (#272)
by reflective recursion on Sun May 25, 2003 at 09:06:29 PM EST

You write...
The reason that people (rightfully) reject the Banach-Tarski paradox on physical grounds is that it relies on certain assumptions that actually don't apply to the physical world. However, mathematicians don't generally care about the physical world

The author of the story writes...
"That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox. But I'd like to point out in this article why this may not be as impossible as one might think at first.
It is not I, who mixed in the word "physical" into this heap of mess. I know what is mathematically going on, but what is metaphorically going on is more than mere mathematical slight-of-hand. A story written for the "layman" which deliberately deceives the target audience is nothing but condescending and elitist.

[ Parent ]
Well (none / 0) (#275)
by Kalani on Sun May 25, 2003 at 10:09:07 PM EST

I know what is mathematically going on, but what is metaphorically going on is more than mere mathematical slight-of-hand. A story written for the "layman" which deliberately deceives the target audience is nothing but condescending and elitist.

I guess I don't see the condescension in the author's approach. I know you weren't saying that you didn't understand the meaning of the paradox but that the author's method of introducing it was misleading and (because he/she assumed that the reader didn't have an intuitive grasp of the complexity of the real numbers and the axiom of choice) elitist. However, I don't think that this is the case. When I read the sentences you quoted, I just see the author preparing the hypothetical layman audience to abandon the requirement that the problem obey physical laws (rather than preparing them to find a trick that makes this all physically possible).

Is it the analogy to doubling the volume and halving the density of physical gasses that bothers you? If that's the case I guess I could agree with you, since that's not exactly how this "paradox" is derived from the axiom of choice (at least not the way I understand it ... the way I understand it is just that the axiom of choice tells us that a set of translations and rotations gets you two spheres of equal volume without telling you how to actually do it).

Still, I think that the author at least provided a great starting point for discussion.

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
No, (none / 0) (#299)
by fhotg on Mon May 26, 2003 at 05:45:01 AM EST

this is wrong. You can keep the idealization of a continous, infinitly divisible physical space and make the "paradoxon" go away anyways. The crux is to use the correct properties of "volume".

[ Parent ]
I thought I acknowledged as much (5.00 / 1) (#312)
by Kalani on Mon May 26, 2003 at 11:10:11 AM EST

From what I've read about the paradox I get the impression that its existence is predicted as a direct result of the axiom of choice. I wasn't suggesting that this is strictly a problem with the idea of continuity (in fact it can be applied to any object isomorphic to a specific group constructed for the problem). Sorry if I gave the wrong impression.

-----
"I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
[ Parent ]
You, sir, are a pedantic jackass [n/t] (2.00 / 2) (#249)
by p3d0 on Sun May 25, 2003 at 03:00:00 PM EST


--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
Sweet, sweet irony! (1.00 / 1) (#273)
by reflective recursion on Sun May 25, 2003 at 09:10:34 PM EST

Tis I, the pedant that criticizes a mathematically essay.

[ Parent ]
You're a real class act (none / 0) (#331)
by p3d0 on Mon May 26, 2003 at 06:32:46 PM EST

I'm glad to see your life is not so busy that you can't take the time to run around K5 modding all my posts as -1.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
Don't forget: I'm a pedant. (1.00 / 1) (#336)
by reflective recursion on Mon May 26, 2003 at 08:31:44 PM EST



[ Parent ]
Naw (none / 0) (#371)
by p3d0 on Tue May 27, 2003 at 08:43:30 PM EST

If you were a real pedant, you would have pointed out that you didn't mod my posts as "-1", but as "1". :-)
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
no, not unless matter has infinite density (none / 0) (#260)
by modmans2ndcoming on Sun May 25, 2003 at 04:12:01 PM EST

which it does not...this paradox is just a mathmatical slieght of hand and nothing more.

[ Parent ]
adjective (none / 0) (#332)
by geozop on Mon May 26, 2003 at 07:27:23 PM EST

It seems your entire arguement comes from the author's use of the word "solid". If he or she had used "whole" or no adjective at all, there would be no problem... And the author could still be trying for the so-called sophism you had accused. Big deal- words shouldn't be held to exact meanings, especially when these numbers are not. *wink*

my handwriting is bad
[ Parent ]
Not even close. (2.00 / 2) (#335)
by reflective recursion on Mon May 26, 2003 at 08:29:38 PM EST

It is not the word "solid" or even as much as the word "physical" (which is a rather key word in my argument) but the whole of his presentation. The front page of K5 has his introduction. In this introduction he sets this paradox as a possible way to duplicate a physical sphere. In short, he deceives his target audience (the layman) so he gains readers. Why does he do this? Simple. Like a number of mathematicians, they derive pleasure and a sense of power from proving how much more knowledgeable they are compared to others. This entire essay is nothing but an ego-boosting parade. Let's look closer...

The intro begins, the author sets the stage by giving the audience a little bit of info...
"Impossible!" I hear you say. "That violates physical laws!" Well, that is what many mathematicians said when they first heard this paradox. But I'd like to point out in this article why this may not be as impossible as one might think at first.
This is written with an implication that there is a way to duplicate a physical sphere, which is utter nonsense. This draws the layman (the unfamiliar) into the article out of interest. If the author was truely honest, not a single person would give a damn. Instead of painting the picture of the infinite and vast, he uses terminology such as "ball" for sphere and mixes in the word "physical" giving the picture a concrete and absolute feel to it.

Once you click and begin to dive in, the author presents you with this:
First of all, let's nail down what exactly we're talking about so that we're all on the same page.
Why, oh why should anyone be on a different page? Afterall, we have only begun reading. There should be no confusion yet. This is the beginning of "I'm smarter than you, look at how much math I know and you don't."

Another clue is the title. "Layman's Guide to the Banach-Tarski Paradox." Now let me ask you this, why would any layman be interested in a hypothetical "paradox" that even mathematicians debate the usefulness or existance of. The simple reason is they don't give a damn. If this were truely an insightful and useful article, then we would leave with mathematical knowledge that would help even the most mathematical ignorant (the "layman"). Instead we are given hypothetical mumbo-jumbo and we are expected to be scratching our heads after reading.

The truely ironic thing is the number of people that don't want me to criticize a mathematical article. Much like you suggest with "words shouldn't be held to exact meanings." If mathematics, itself, was criticized as loosely as you and others wished, it would have all toppled over long ago.

[ Parent ]
well (none / 0) (#378)
by geozop on Thu May 29, 2003 at 05:58:41 AM EST

You're not even critizing the thoery or its meaning, but rather, the author for trying to show something a bit unusual. Perhaps even try to start a discussion about the axiom. I never thought that any kuro5hin reader would imagine that he or she could double an inventory of balls with a knife. Anyone who tredged through the beginning of the article would already know it's a mathmatical trick, not something physically possible.

Additionally, did you miss the lines he used in the article: "Obviously, this is impossible with a physical sphere; ..." And the second sentence: "First and foremost, we're talking about a mathematical sphere, ..." He's already telling the tricked readers that it's not physically possible.

I don't see why you are bothering your time with something that could only be a shenanigan to feed an ego, disguised as a math problem. There are many other like problems in the world, most of which deceive people in order for real power or money. I believe that they deserve your attention, since you seem to have the energy to think intelligently and also attack.

my handwriting is bad
[ Parent ]
Clinching Proof of the Non-Existence of God (2.00 / 2) (#199)
by marktaw on Sun May 25, 2003 at 04:23:00 AM EST

Now it is such a bizarrely improbable coincidence that anything so mindboggingly useful could have evolved purely by chance that some thinkers have chosen to see it as the final and clinching proof of the non-existence of God.

The argument goes something like this: `I refuse to prove that I exist,'says God, `for proof denies faith, and without faith I am nothing.'

`But,' says Man, `The Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED.'

`Oh dear,' says God, `I hadn't thought of that,' and promptly vanished in a puff of logic.

`Oh, that was easy,' says Man, and for an encore goes on to prove that black is white and gets himself killed on the next zebra crossing

- Douglas Adams, I think we all know where from.

My Infinitely Hidden Comment (1.50 / 2) (#200)
by marktaw on Sun May 25, 2003 at 04:33:07 AM EST

I hid this one underneath the funny one.

So if you start with an infinite set of numbers and cut it in half you still have an infinite set of numbers. BIG DEAL! I learned this in high school.

There are an infinite series of numbers between 0 and 1. There are also an infinite series of numbers between 0 and 0.5.

Guess what, there are infinite sets that are smaller than other infinite sets. This is (I believe) a generally accepted mathematical principal. This whole sphere hype thing is just silly.

[ Parent ]

Really? (5.00 / 4) (#203)
by i on Sun May 25, 2003 at 06:04:50 AM EST

Then prove an analogous theorem for two dimensions, genius.

and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]
How is it different? (none / 0) (#264)
by Eater on Sun May 25, 2003 at 05:06:37 PM EST

How is a set of two-dimensional points any different that a set of three-dimensional points in this context? Just replace "sphere" with "circle" in the above explanation.

Eater.

[ Parent ]
The problem with that. (4.66 / 3) (#270)
by i on Sun May 25, 2003 at 06:46:04 PM EST

What we have above is an explanation, not a proof. Replacing 3D with 2D in it will again yield an explanation, not a proof.

There is a proof for the 3D case. I linked it in another comment, but here we go again. Can you convert it to a proof for the 2D case?

Hint: not bloody likely.

The free group F on two generators is  F-paradoxical, whereas the free group Z on one generator (a.k.a. "the integers") is not Z-paradoxical. Rougly, if G acts on S, S is said to be G-paradoxical if Banach-Tarsky decomposition exists for S under transforms from G.

The proof is centered around picking a special subset of all rotations of the sphere. This subset turns out to be just F. If you do it for a circle you get at most Z. Pity, isn't it?

and we have a contradicton according to our assumptions and the factor theorem

[ Parent ]

In that case, the explanation presented... (4.00 / 1) (#288)
by Eater on Mon May 26, 2003 at 02:47:22 AM EST

...by the article is obviously lacking something, because it fails to explain why this applies specifically to a 3-dimensional object and not a 2 or N-dimensional one. The most the article explains is that "copies" of the sphere can be made because it is infinitely divisible. But thanks for the information, it did indeed help me understand this thing a bit more.

Eater.

[ Parent ]
Wait a minute (none / 0) (#290)
by marktaw on Mon May 26, 2003 at 04:36:34 AM EST

I've looked at http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf and it seems that this post is a somewhat of a distillation of that article.

On message boards and in newsgroups I've seen a lot of "do my homework for me" posts. They go something like this: I have a project where I have to __, does anyone have any idea what I should do?

Is this post a variation on that? It is finals time, is this a this is my final report, please critique it so I can improve it before I hand it in post?

As far as proving it for two dimensions, I can't do it with numbers and variables, and I don't feel like actually hashing through all the formulas required for even the 3 dimensional proof, but allow me to commit the same 2 dimensional sin Edwin Abbot commited in Flatland.

Imagine a disc of infinite surface area. This is a fluid surface, sort of like the surface of a lake.

If you slice it into four pieces, and remove two of them - any four pieces, nevermind whether they intersect the center or not, the remaining two pieces will flow back into the same surface area. Why? Because the remaining infinite surface area of the lake gladly fills in the infinite space of the original disc.

Your math teacher would probably fail me for saying that.

[ Parent ]

you get an F for that problem (none / 0) (#300)
by martingale on Mon May 26, 2003 at 06:16:42 AM EST

There's your problem, right there: Imagine a disc of infinite surface area.

Sorry, can't work. Where do you put the elephants?

[ Parent ]

C'mon we all know that Earth is really flat (none / 0) (#365)
by marktaw on Tue May 27, 2003 at 03:03:12 PM EST

And that the moon landing was faked on the back lot at Paramount.

[ Parent ]
The poll misses an important option (3.33 / 3) (#212)
by jope on Sun May 25, 2003 at 07:37:33 AM EST

Namely: the paradox makes sense, but the rationalization doesn't. It might help to convince people to accept the fact that such a paradox can exist, but it does it on entirely mislieading grounds. Rather than trying to grasp infinity-related things by wrong analogues, one should try to learn accepting the non-graspability of the concept, IMO. This helps avoiding your intuition when it would do harm.

The concept is graspable... Now. ;-) (3.66 / 3) (#303)
by Viliam Bur on Mon May 26, 2003 at 07:43:39 AM EST

I did math for a couple of years, but since the high school it is not possible for me to understand the details of all fields. The BT paradox made me nervous, because I use my intuition whenever possible when doing math - and I could not imagine this one.

Of course, in math you have to prove everything mathematically... but it is often the intuition that shows you the direction for the proof... and the rest is only the translation into math terms, with getting some ideas more exact.

I do not have sufficient background to understand an explanation of BT if it were written by some equations and Greek letters. However, now it is pretty possible that I could (after reading a book or two to get used to terms and definitions) write a proof myself. OK, probably I am only dreaming now. ;-)

In theory, you do not need imagination, the equations should be enough. In practice, imaginations and analogies are strongly compressed information - and yeah, the compression algorithm is not errorproof... you have to check the results later - but it still helps.

[ Parent ]

Excellent article (1.33 / 6) (#228)
by dmt on Sun May 25, 2003 at 11:28:06 AM EST

It all comes back to the nature of infinity, it's relation to the physical world and the axiom of choice.  Which is a bloody great grey area in an infinite sea of grey areas.  Yours is as good an explanation of the Banach-Tarski Paradox as any I've heard.

I think one of the things that puts people off this kind of math is the terminology.  It really is quite alien to those used to more procedural ways of thinking; I think it would be great if someone such as yourself could produce a laymans guide to mathematical terminology, juxtaposed with more procedural examples of thinking.  Although when I was studying my mathematical thinking wasn't really any good until I began to program in ML, then things just fitted into place because I could visualize them.

[n/t] ZERO RATE THE ABOVE IT IS A DUP - SORRY! (5.00 / 1) (#233)
by dmt on Sun May 25, 2003 at 12:47:57 PM EST



[ Parent ]
Excellent article (none / 0) (#229)
by dmt on Sun May 25, 2003 at 11:32:26 AM EST

It all comes back to the nature of infinity, it's relation to the physical world and the axiom of choice.  Which is a bloody great grey area in an infinite sea of grey areas.  Yours is as good an explanation of the Banach-Tarski Paradox as any I've heard.

I think one of the things that puts people off this kind of math is the terminology.  It really is quite alien to those used to more procedural ways of thinking; I think it would be great if someone such as yourself could produce a laymans guide to mathematical terminology, juxtaposed with more procedural examples of thinking.  Although when I was studying my mathematical thinking wasn't really any good until I began to program in ML, then things just fitted into place and I could visualize.  

Of course ML is functional language, so maybe explaining things in procedural terms is a red herring.  But I dare say more program in procedural and (semi) OO languages than ML.

Math terminology (4.00 / 1) (#247)
by The Writer on Sun May 25, 2003 at 02:51:27 PM EST

As with all terminology, math terminology exists because mathematicians need precise terms with precise meanings in order to convey what they want to say. Of course, the terminology would be drawn from the closest words in natural language, which can become confusing to non-mathematicians because they understand those words differently. There's always the need for clarification when you use terms with people who aren't familiar with your definition of it.

As for explaining things in procedural terms... some things just cannot be explained that way, unfortunately. The procedural bias is the source of such incorrect statements as the "repeat X forever and you get Y" fallacy I've pointed out in several other comments. It is more dangerous in this case, because the seemingly easier-to-understand procedural description gives people a false impression that they understand the concept, when they really don't. This then leads to contradictions and inconsistencies which they attribute to mathemetics; but actually the contradiction comes from the (incorrect) procedural description, not from a real contradiction in the math.

[ Parent ]

Makes sense. (5.00 / 2) (#253)
by dmt on Sun May 25, 2003 at 03:19:20 PM EST

I suppose a Math through Quickbasic article is out of the question?  I jest ;-).

However, something along similar lines; a 'debunking common conceptual errors' article would be really useful for a lot of people.  I'm sure that those of us who have worked with higher-level math, but don't do it professionally, would benefit from having errors that maybe we make, but we're unaware of pointed out.  

The example you point out is a prime example.  I guess the problem would be finding out what are the most common conceptual errors and among whom...

[ Parent ]

Good idea (5.00 / 2) (#257)
by The Writer on Sun May 25, 2003 at 03:38:32 PM EST

I think a series of articles debunking common misconceptions would be great, and within my capability. (After all, I'm not a mathematician by profession.)

[ Parent ]

Am I really? (none / 0) (#414)
by vile on Thu Jun 26, 2003 at 04:58:07 AM EST

That light? Nope.

~
The money is in the treatment, not the cure.
[ Parent ]
terminology (none / 0) (#298)
by fhotg on Mon May 26, 2003 at 05:34:25 AM EST

Nearly all mathematicians have green hair.

[ Parent ]
Why 5 pieces? (5.00 / 2) (#234)
by zml on Sun May 25, 2003 at 12:50:14 PM EST

Okay, I understand your density argument, and I come from a reasonable mathematical background (it's just all getting lost over the years). I've even attempted to read deeper into this subject to see the proof of my question, but I really would like a layman's (or perhaps a mid-level) answer here:

Why does it require 5 pieces to accomplish this? By the same principal as the odd/even split, it seems like you should be able to do this in 1 "cut" (since your sets seem to have no bearing on a physical "cut"). Since a sphere is a closed neighborhood of sorts around the center, you should be able to just extract half the points of the set, translate that "half" subset away from the original sphere, and end up with two closed neighborhoods around different centers, both constituting a sphere.

I think the answer to my question may lay in the fact that this set is not guaranteed to be constructible. But I'm not sure you need that to show it's possible, you should be able to get away with saying "well, this subset exists, regardless of whether I can show you how to construct it" (but there's probably some fallacy there, too).

Thanks for getting my math juices flowing again..

A layman's guess (3.00 / 2) (#248)
by p3d0 on Sun May 25, 2003 at 02:57:06 PM EST

Let's ignore the point in the center of the sphere. That leaves four pieces. Why four? Consider this...

Remember that the paradox says we can throw away half the sphere, and what we're left with can be reconfigured back into the original sphere. So, oversimplifying a bit, supposing we start out with 2N pieces, throwing away half the sphere would leave us with N. Those N can be reconfigured into a sphere.

Well, if N=1, then there's no "reconfiguring" to be done: that one piece just is what it is, and (I presume) it's not a sphere, so no amount of translation or rotation will make it a sphere. The smallest number of pieces that can be "reconfigured" at all is 2, and so we must start out with at least 4 pieces.

Pretty hand-wavy, I know, but that's the best I can do.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]

I think it works better with one. (2.00 / 1) (#291)
by marktaw on Mon May 26, 2003 at 04:41:42 AM EST

Watch as I turn two spheres into one.

1 sphere multiplied by 1 sphere = 1 sphere.

Spooky isn't it.

Sorry, I'm in a facetious mood.

[ Parent ]

There's nothing spooky about it (none / 0) (#297)
by synaesthesia on Mon May 26, 2003 at 05:33:44 AM EST

One sphere multiplied by one sphere = one sphere2.

Sausages or cheese?
[ Parent ]
No. (4.00 / 1) (#334)
by Canar on Mon May 26, 2003 at 07:55:14 PM EST

There is definitely something spooky there.

Honestly, have you heard of a square sphere?

[ Parent ]

surely there's a simple answer to this.... (4.00 / 1) (#235)
by romperstomper on Sun May 25, 2003 at 12:52:07 PM EST

Ok, not to be a complete idiot, but my question concerns the part of the article that says a physical sphere, unlike a mathematical sphere, is "finite".  

It certainly seems logical, but I'm wondering... to say that a physical sphere is finite, aren't you saying that it can be broken down into a smallest component of some sort.  That is to say that you could cut it in half over and over again until you finally declared what you have to be the smallest component of the sphere.  And if you can do that, what defines the smallest piece?  (atoms, quarks, the goo inside??)

Or is "finite" a term associated more with some other form of measurement?  

I'm also wondering if it's possible (theoretically) to assume that a real world object could be infinitely small (meaning that perhaps atoms could be broken down into billions of smaller pieces, and then those pieces into more, and so on into infinity)

It seems that it's comparable to theories that the universe is infinitely large.  Is the theory that it's infinitely small in some way more radical?

Now, everyone rush to post an answer that will remind me why I should wait for someone else to ask my questions and be ridiculed!

What "finite" means (3.00 / 2) (#245)
by p3d0 on Sun May 25, 2003 at 02:49:32 PM EST

In this context, I think "finite" simply means that if you divide a physical sphere small enough, it stops acting like a mathematical sphere.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
it is impossable to have a quark by itself (none / 0) (#259)
by modmans2ndcoming on Sun May 25, 2003 at 04:09:20 PM EST

Quarks only exist in triplets, and many people think of them more as a way to discribe an abstract idea rather than a real thing.

atoms are considered the smalles peice of actual matter, after than you enter the realm of quantum physics where matter is energy and energy is matter.

[ Parent ]

Mesons... (5.00 / 3) (#282)
by kerinsky on Mon May 26, 2003 at 12:21:39 AM EST

Baryons (ie Nucleons and Hyperons) are made of up 3 quarks.  Mesons are made up of paired quarks however.  See this chart, it's old, but gets the point across.  There is also this.

-=-
A conclusion is simply the place where you got tired of thinking.
[ Parent ]
It's true (5.00 / 1) (#301)
by Three Pi Mesons on Mon May 26, 2003 at 06:33:56 AM EST

what he says about mesons. A pi-meson, or "pion" (not "peon", please!) is made of either
  • up and anti-down
  • anti-up and down
  • a superposition of (up and anti-up) with (down and anti-down)
That makes three kinds of pi-meson. Dali thinks we look like this.

:: "Every problem in the world can be fixed with either flowers, or duct tape, or both." - illuzion
[ Parent ]
thank-you for the correction... (none / 0) (#325)
by modmans2ndcoming on Mon May 26, 2003 at 03:03:26 PM EST

but Quarks do not exist out in the wild as a bare quark...hence my statment that they are used more as a way to describe features of elementry particles rather than actualy being elementary particles

[ Parent ]
in this context (none / 0) (#294)
by fhotg on Mon May 26, 2003 at 05:16:54 AM EST

it is not necessary to contemplate the real (physical) structure of matter, i.e. atoms, quarks, whatever. You get the phenomenon "in line" by honoring the fact that not any arbitrary subset of R^3 describes a part of the physical space. How big that subset is, is not important. It's not a matter of "getting too small". It's more that there is no measure for the bigness of these freak sets defineable in the first place.

This paradox is still one if you double a sphere of vacuum, no matter involved.

[ Parent ]

just a silly question (5.00 / 1) (#250)
by welkin on Sun May 25, 2003 at 03:06:11 PM EST

For this paradox, could one only create a finite number of spheres?

Yes and no. (none / 0) (#265)
by pla on Sun May 25, 2003 at 05:20:30 PM EST

Not in one step, since the heart of this "trick" involves the fact that infinity divided by any finite number yields infinity. If you try to make an infinite number of subdivisions, you get... well, something messy, but arguably "1".

But you can repeat the "split one into two of the same size" an infinite number of times.

Remind me again how theoretical mathematicians help us describe reality? ;-)


[ Parent ]
Infinities.. (5.00 / 1) (#281)
by zml on Sun May 25, 2003 at 11:49:42 PM EST

You're mixing infinities and probably need some clarification here.

It would be possible to make a countably infinite number of cuts in the sphere and still have enough points in each cut to make a new sphere. For instance, the real number line can be partitioned into mixed open/closed intervals, like [0,1), [1,2), etc.

Countably infinite means that it is equivalent to the number of natural numbers (0,1,2,3, etc.).

The reals are *not* countably infinite. The proof for this is essentially to assume that there is some mapping from the natural numbers to the reals, then show that you can form yet another real number that doesn't fit the mapping.

I'm not sure whether the Banach-Tarsaki paradox applies to a countably infinite number of cuts. I'dactually wager that it does, but it's an interesting question. However, I'm guessing it does not apply to an uncountable number of cuts.

Your second comment is interesting. In the mathematical world, it is not necessarily the case that if you can do <x> for a finite number of times and property <y> still holds, then you can do <x> for an infinite number of times and property <y> still holds. Most people think this intuitive (well, if you can keep doing it, why doesn't it hold over the infinite case?), but it's not necessarily true. One example is a variant on the power set, the n-tuple mapping. You can show that you can map (x,y) for integer x,y onto unique integers (thereby showing that the cardinality of tuples is equal to the cardinality of the integers). Similar, you can show that (x,y,z) maps, and that (a,b,c,d) maps, and that (a1,a2,a3,...,an) maps for all n. But it breaks down in the infinite case. You cannot take an infinitely long string of numbers and map it back to the integers.

A lot of things break down at infinity..

[ Parent ]

Things that break down at infinity (5.00 / 1) (#370)
by Frigorific on Tue May 27, 2003 at 07:06:18 PM EST

Another example is the basic integral from calculus. It's pretty easy to prove that you can remove finitely many points from the function and still obtain the same value of the integral, but you can't take out an infinite number of points--the set of points you take out has to have measure zero.

-Brendan



[ Parent ]
just an intuitive answer (2.00 / 3) (#302)
by Viliam Bur on Mon May 26, 2003 at 07:27:19 AM EST

If you split one (finite) sphere into finitely many pieces, you cannot make infinite number of spheres by rotation and translation. Why? Imagine that to each piece is attached the boundary of the original sphere.

As you move and rotate the pieces, the boundaries are only moved (because sphere rotated is just another sphere of the same size, perhaps also moved - if you do not rotate around its center). So at the end you have N pieces moved and rotated, which are inside of N spherical boundaries - they cannot make more than N spheres.

If you cut one sphere into infinitely many pieces... then probably you can get infinitely many pieces. But I am not completely sure about that. The solution "cut into 4 pieces and make 2 spheres, then cut again,... and now do all this in one step" is not valid, because the number of pieces would grow exponentially... which would be more than a countable infinite. However, there may be a smarter solution of cutting into infinitely many pieces in one step and then from each 2 of them making one sphere.

[ Parent ]

You can, recursively (none / 0) (#339)
by hugues on Tue May 27, 2003 at 02:00:23 AM EST

If you can make 2 spheres by the BT method, and if these spheres are indistinguishible, which they are, then you can make 4 spheres, 8, 16, etc. As many as you want, eventually countably infinitely many spheres.

[ Parent ]
Repeated doublings don't yield infinity (none / 0) (#358)
by rujith on Tue May 27, 2003 at 12:31:06 PM EST

As many as you want, eventually countably infinitely many spheres.

That's not the same thing: given any positive integer N, you can get that many spheres by repeated applications; but you can't get an infinite number of spheres.

[ Parent ]

A few questions (5.00 / 1) (#256)
by salsaman on Sun May 25, 2003 at 03:36:36 PM EST

Is there only one solution to this, or are there a number of ways to make the cuts ?

Also, does this only work for 3spheres, or can it be applied to Nspheres generally ?

Finally, does it only apply to the sphere, or could there be an equivalent for other shapes, e.g. a cube or a torus ?

If I understand correctly... (5.00 / 1) (#262)
by Eater on Sun May 25, 2003 at 04:59:47 PM EST

The shape doesn't matter, as long as it's a set of points that is infinitely divisible (it can even be in 2, 1, or N dimensions). As for the way to make the cuts, they are not really cuts, because every "part" that you "cut out" has to be infinitely complex... but I believe there are many different ways to make the "correct" cuts, as long as the resulting parts are infinitely complex.

Eater.

[ Parent ]
Dimensions (none / 0) (#344)
by cep on Tue May 27, 2003 at 07:44:29 AM EST

No, it is proven that there is no Banach-Tarski-Paradox in 1 or 2 dimensions. I have read this in the book by Stan Wagon abot the paradox, so I can't tell you about the proof. But the actual construction of the dissection needs rotations around two different axes, which cannot be generalized to less than 3 dimensions.

For more than 3 dimensions, there should be a Banach-Tarski-dissection. A higher-dimensional sphere is a kind of "stack" of 3-dimensional spheres - the same way a 3-sphere is a "stack of circles", like infinitely thin book pages - so you should be able to get a Banach-Tarski-decomposition of a higher-dimensional sphere by dissecting each of its 3-dimensional slices.

[ Parent ]

How to make the cuts (none / 0) (#381)
by bwcbwc on Thu May 29, 2003 at 03:28:28 PM EST

Not sure about the nitty gritty, but I bet some of the cuts have to involve translating irrational numbers (uncountably infinite set with positive Lebesque measure) into rational numbers (countably infinite with Lebesque measure zero). For example, take irrationals m*n/(k*pi), where k is an integer, m and n are positive integers, and translate the set by multiplying by pi/n, and you get the set of rational numbers.

Also, I'm OK with the axiom of choice for a countably infinite set of sets, but an uncountable set of sets??? How do you "iterate" over that?

[ Parent ]

Number of possible cuts (none / 0) (#384)
by The Writer on Thu May 29, 2003 at 05:06:54 PM EST

There are an unbounded number of possible cuts, AFAIK. The 5 pieces is the minimal number required for the paradox. One feature about the cuts is that at least one of the pieces must have no Lebesgue measure.

[ Parent ]

just before Igot to Bingo!! (4.50 / 2) (#258)
by modmans2ndcoming on Sun May 25, 2003 at 04:05:39 PM EST

I was thinking that the reason you get double the mass is becasue you have and infinitly divisable object....what is infinity divided by 2? infinity.

I totaly agree with this conclusion...it is more of a mathmatical slight of hand.

another real life example (4.75 / 4) (#278)
by urdine on Sun May 25, 2003 at 10:36:23 PM EST

The point of the rotating and translation seems to be to get around the ugly fact that moving a specific point out of the source sphere must necessarily leave a void... which could only be filled by translating a point from somewhere else.  Which of course leaves us another void... which could be filled with a point from somewhere else, and on and on it goes like an Escher painting.

Anyway, I think a simpler way to visualize the paradox is to imagine a glass sphere filled with an infinitely dense gas.  If you fill a second sphere with gas from the first sphere, both spheres will then be filled with gas, the ugly translation and rotation taking place on a micro-scale, like it would with real gas.

Excellent article! (2.16 / 12) (#287)
by AmberEyes on Mon May 26, 2003 at 02:03:18 AM EST

I'm looking forward to your follow-up article in which you explain the Mental-Masturbation Paradox; namely, how you can take something that can only exist in imaginary-world-land and has as much relevance as a negative square root, and think you're doing something productive by writing about it like you think it's real or something.

-AmberEyes


"But you [AmberEyes] have never admitted defeat your entire life, so why should you start now. It seems the only perfect human being since Jesus Christ himself is in our presence." -my Uncle Dean
yeah, imaginary numbers (5.00 / 4) (#293)
by fhotg on Mon May 26, 2003 at 05:06:45 AM EST

have been shown to be completely useless, particularly in electrical engineering. Common sense rulez.

[ Parent ]
Hello smart person. (none / 0) (#309)
by tkatchev on Mon May 26, 2003 at 09:30:40 AM EST

Clue: what you call "imaginary numbers" in electrical engineering is really a two-dimensional vector in R^2.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Hello engineer (none / 0) (#318)
by fhotg on Mon May 26, 2003 at 01:14:21 PM EST

What you mean that ressembles R^2 is the field of the complex numbers. Since those describe much more elegantly the properties we're interested in than the vector which fits better your imagination, the latter is spurious.

You divide vector by vector on a regular basis ?

You can show in 5 lines that e^(i*pi)+1=0 ?

[ Parent ]

Your point is spurious. (none / 0) (#321)
by tkatchev on Mon May 26, 2003 at 01:28:42 PM EST

It's irrelevant whether 2-D points are described as "x + iy" or "xi + yj".

Simply define "vector division" to match "complex division", and you're set.

"Complex numbers" are just 2-D vectors that have a couple extra properties defined for them in order to fool you into thinking that they are "almost the same as reals".

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

if you have no other way (none / 0) (#322)
by fhotg on Mon May 26, 2003 at 02:10:28 PM EST

to grasp it.

So be it. But better don't write this in you next term-paper.

[ Parent ]

Heh. (none / 0) (#326)
by tkatchev on Mon May 26, 2003 at 04:04:01 PM EST

I'm a diplomated mathematician, so don't argue with me.

Fact is, "imaginary numbers" are simply points in R^2 with a couple extra properties defined for completeness' sake.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Don't be a jackass (5.00 / 2) (#338)
by EggplantMan on Tue May 27, 2003 at 12:23:21 AM EST

First, Mr. Mathematician, imaginary numbers are not points in R2. They are points on the imaginary number line. I think you were talking about complex numbers. Complex numbers aren't just 'points in the plane' either though, any more than Pn are points in an n+1 dimensional vector space. It just so happens that complex numbers are a 2D vector space and R2 is 2D also, so they are isomorphic. I think what you're trying to say is that they're isomorphic. That doesn't mean that they're the same thing.

[ Parent ]
Yes it does. (none / 0) (#343)
by tkatchev on Tue May 27, 2003 at 06:23:23 AM EST

"Isomorphic" translated from mathematese means "the same thing".

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Erm, no. (none / 0) (#345)
by DrH0ffm4n on Tue May 27, 2003 at 10:16:51 AM EST

Isomorphic means "having the same form". It is variously used in different branches of maths, but rarely means "the same thing".

e.g. The set of naturals is isomorphic to the set of even numbers,
but they are not "the same thing".

Maybe I'm being a pedant. I believe what you meant is that they can be used interchangeably in this context.


---
The face of a child can say it all, especially the mouth part of the face.

[ Parent ]

Actually, my real point... (5.00 / 1) (#360)
by tkatchev on Tue May 27, 2003 at 01:16:14 PM EST

...is that it is counterproductive to throw around phrases like "the same thing" when you don't know precisely what you mean, especially in mathematics.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Whatever (none / 0) (#401)
by p3d0 on Mon Jun 02, 2003 at 08:45:54 AM EST

Complex numbers are a useful and elegant way to solve certain real-world problems. End of story.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
What are you, some kind of puritan? (2.00 / 2) (#296)
by delmoi on Mon May 26, 2003 at 05:22:33 AM EST

What's wrong with masterbation?
--
"'argumentation' is not a word, idiot." -- thelizman
[ Parent ]
Negative numbers (none / 0) (#400)
by p3d0 on Mon Jun 02, 2003 at 08:42:14 AM EST

Why are imaginary numbers any less useful than negative numbers?
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
A really nice piece (4.50 / 2) (#289)
by fhotg on Mon May 26, 2003 at 03:48:14 AM EST

When I was contemplating about how to "marry the B-T paradox to the intuition", I came to the conclusion that at the root of the paradox lies the confusion of physical space with ℜ3. It is very common to just equate the two, but this paradox shows that physical space has a more demanding structure and needs to be considered with a Borel - Sigma - Algebra if you want to use the idea of "volume".

points (2.00 / 3) (#292)
by relief on Mon May 26, 2003 at 05:00:53 AM EST

this mathematical blip is due to the lack of understanding of the "point" in "space". it really just depends on how one defines things. its nothing groundbreaking. if one has the time to ponder about this "paradox", i'd recommend first getting the fundamentals, like whether two adjacent points in a continuum is the same point or not.

this is why i don't like *some* math. what a waste of brain.

----------------------------
If you're afraid of eating chicken wings with my dick cheese as a condiment, you're a wuss.

"Two adjacent points in a continuum"? (5.00 / 2) (#308)
by tkatchev on Mon May 26, 2003 at 09:29:23 AM EST

Whatever do you mean by that?

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

math v physics (none / 0) (#310)
by bauklo on Mon May 26, 2003 at 09:47:28 AM EST

i am not the author of that comment, but what i would suggest he meant is: if you have 2 "points" (having the size 0 in all directions)in a space, which are next to each other,so that the distance is infinitively small, is it actually the same point or not? mathematically speaking probably not, but physically probably yes.

[ Parent ]
the point was: "next to each other"... (5.00 / 2) (#317)
by warrax on Mon May 26, 2003 at 01:04:47 PM EST

... does not make sense because if you have two points a distance d apart (in R^n), there is a point between them that is d/2 from either of them. Thus the original points were not "next to" each other originally. This is a contradiction, so the original assumption that the points were "next to each other" is false. QED.

-- "Guns don't kill people. I kill people."
[ Parent ]
Well... (none / 0) (#319)
by tkatchev on Mon May 26, 2003 at 01:24:42 PM EST

Unless you mean something like "two points x_1 and x_2 such that for all epsilon > 0 |x_1 - x_2| <epsilon". <P> This is why I asked -- honestly, I have absolutely no idea what the poster meant by "next to each other". I asked out of simple curiosity.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Well... (none / 0) (#320)
by tkatchev on Mon May 26, 2003 at 01:25:39 PM EST

Unless you mean something like "two points x_1 and x_2 such that for all epsilon > 0, |x_1 - x_2| < epsilon".

This is why I asked -- honestly, I have absolutely no idea what the poster meant by "next to each other". I asked out of simple curiosity.

   -- Signed, Lev Andropoff, cosmonaut.
[ Parent ]

Oh, ok. Sorry... (none / 0) (#324)
by warrax on Mon May 26, 2003 at 02:51:34 PM EST

... just attempting to clarify that there is really no such thing as "next to each other" in a continuum. That two points, x and y, are "next to each other" usually means that there is nothing in between x and y. But on the real axis there is always a point (infinitely many, in fact) between x and y (assuming that x!=y). So it doesn't really make sense to talk about "next to each other" for the reals.

-- "Guns don't kill people. I kill people."
[ Parent ]
Two adjacent points (5.00 / 2) (#314)
by Kinthelt on Mon May 26, 2003 at 12:21:32 PM EST

If you have a continuum, you cannot have two "adjacent" points. At least not in the canonical sense. You *can* have two points less than a certain distance (however you want to define distance) apart. But you will always have an infinite number of points between any two points unless they are the same point.

[ Parent ]
Identical? (2.33 / 3) (#307)
by scart on Mon May 26, 2003 at 09:12:16 AM EST

Basically, you are saying that you take an infinately large set of unique elements, devide all the elements between two other sets, then the resultant sets will be identical to each other, as well as the first set. I'm sorry, but that just does not work. Any element in the original set will only be present in one of the result sets, and the two result sets will have no elements in common.

Using infinately divisable sets does not help you at all. No matter how finely you devide the base for your set, no two elements will ever be equal to each other.

Your result sets might have the same characteristics, ie. infinate number of elements, same distribution pattern, same 'density', but they will never be identical.

Different infinities (none / 0) (#327)
by Homburg on Mon May 26, 2003 at 05:41:38 PM EST

I was going to say I didn't think your reasoning applied to sets which are not just infinitely large, but also infinitely dense (i.e., the real numbers or  three-dimensional space).

Consider the numbers n and n + e, where e < r for all r in R+. Then you have the sets S1 = R - {n}, and S2 = R - {n + e}. Are these different sets? I'm not sure; it depends on the subtlety of comparing sets with non-denumerably many elements, and that's something I'm not sure about.

Intuitively, though, this looks a lot like saying S1 and S2 are different sets, but have the same members, or have different members but are identical, or something equally incoherent.

Anyone who knows more about the subject care to explain this?

[ Parent ]

Welcome to Infinite Sets (5.00 / 3) (#349)
by hal200 on Tue May 27, 2003 at 10:46:44 AM EST

It's mindbending, but you're both right.  Let me explain.

What you're saying, in mathematical terms, is that if you take an inifinite set, X with a cardinality of C, and split it into two disjoint subsets, A and B, then neither A nor B are equivalent to X.  This is true. Give yourself a cookie. :)

However, if you read the article again, you'll note that an important part of the process is applying a transformation function to the subsets.  At that point, all bets are off.  

Take the given example of the set of all positive integers {0,1,2,3,...}, split into two subsets.  E = {0,2,4,6,...} is the set of all even numbers  (plus 0) and F = {1,3,5,7...} is the set of all odd numbers.  We're going to focus of E here, but the same idea could be applied to F.

So, we have N, which is the infinite set of all positive integers.  We've split it into two infinite subsets, E and F.  E is is comprised of 0 and all the even numbers, and is most certainly not equivalent to N.  The proof of this is dead simple, and intuitive.  All you have to do is find an element that belongs to N, but not in E.  Well, any of the elements in F fit the bill.  Still with me so far?

Now here is where the magic comes in.  We define a transformation function T(y), and apply it to every element of the set E, taking the results and collecting them up into set E'.  In this case, T(y) = y/2.  Because we know that all the elements in E are even, we know they are all evenly divisible by 2.  

So, what does that leave us with?  Well, 0/2=0, 2/2=1, 4/2=2, 6/2=3 and so on and so on...So, E' = {0,1,2,3,....}, which is EXACTLY the definition of our originial set of N.  Wierd, no?

What will really bake your noodle is that the integers are what is known as a countable set. Any countable set can be made equivalent to any other countable set.  Thus, with the right transformation function, the set of positive integers can become equivalent to the set of all integers.  

However, there are also the uncountable infinite sets, such as the real numbers, which are proveably larger than the countable infinite sets.  In other words, it's impossible to assign a unique integer to every possible real number.  For example, if you assign the number 1 to 1.0, and 2 to 1.1, what do you do about 1.01?  And if you consider 1.01 to be 2, what about 1.001?

Whew! That got a little longwinded there! Hopefully that clarifies things for ya. (Or at least convinces you to stay away from university level Math if you value your sanity. ;)


[ Parent ]

I'm confused... (none / 0) (#402)
by DavidTC on Mon Jun 02, 2003 at 11:30:16 AM EST

...as to the point of this.

If you have an infinite amount of points, and can move them around and do tranformations on them, can't you make anything and everything?

I mean, every single set of mathmatical objects consists of one infinite set of points, no matter the size or shape.

Or is there some special meaning of 'transform' and 'rotate' I'm not getting?

-David T. C.
Yes, my email address is real.
[ Parent ]

The answer is YES BUT... (none / 0) (#403)
by TuringTest2002 on Mon Jun 02, 2003 at 01:28:02 PM EST

..you probably would need an infinitely complex transformation function (i.e. an algorithm with infinite steps), so it would'nt be much useful in a practical sense.

It's not only that mathematical objects exist, it's about you being able to find them! ;-)

[ Parent ]

Uh, no (5.00 / 1) (#405)
by The Writer on Mon Jun 02, 2003 at 03:33:14 PM EST

The algorithm doesn't matter.

Cantor proved that it was impossible to assign an integer to every real, and the way he proved it was by assuming that it was possible, and then deriving a contradiction. If there were as many reals as naturals, then it must be possible to have a 1-to-1 correspondence between every real to a unique natural. You could then put the reals in a sequence, by which natural number they corresponded with. So you have a list L={x1, x2, x3, ...} where x1 is some real number corresponding to 1, x2 is a real corresponding to 2, x3 to 3, and so on. We don't really care if there's an algorithm to do this; the point is that we take every real and put them in a list, and label them 1, 2, 3, and so on. If we could do this, then we could, indeed, say that there are as many reals as there are naturals, since we just found a 1-to-1 correspondence between them.

However, the problem is that L does not contain every possible real. Here's why: write x1, x2, x3, ... in decimal form. Eg., 32.312423..., 0.392540..., and so on. Most of these numbers will have an infinite number of decimals, since most reals aren't integers. So far so good. Now we construct a Pathological Number, P, as follows: for simplicity, let's just say the integer part of P is the same as x1. In this case, that's 32. Now, for the first digit after the decimal point, we choose something different from 3, which is what x1 has. Say we choose 4. So P=32.4..., which makes it not equal to x1.

For the next digit, we choose it to be different from the 2nd digit of x2. Say we choose 0 (since the 2nd digit of x2 after the decimal is 9). This makes P not equal to x2 either. We continue this way, making the n'th decimal digit of P different from the n'th decimal digit of xn.

I leave it to you to conclude that P cannot be in the list L. (It mismatches every number in the list L in at least one digit, so it cannot be equal to any of them.)

But wait, there's a contradiction: P is certainly a valid real number, but it is not in L, which we assumed contains a list of every possible real number.

Conclusion: it is not possible to put the real numbers into a 1-to-1 correspondence with the natural numbers, no matter how clever your algorithm. That is to say, the cardinality (size) of the set of reals is not the same as the cardinality of the set of natural numbers. Furthermore, we know that there are at least as many reals as naturals, since the real numbers are a superset of the natural numbers.

Therefore, there must be more reals than naturals. Even though both sets are infinite, one is "more infinite" than the other.

This proof may seem rather simple (with a little twist, perhaps); but it has vast consequences. One consequence is that you cannot rearrange the natural numbers to form the reals, no matter how you try to re-arrange them. There are just "not enough" naturals to cover every possible real. The proof of this is quite simple, actually; it's just the mirror-image of the proof I gave above. Even though you have infinitely many naturals to work with, and you are also permitted to perform infinitely complex operations on them (as long as you don't introduce new objects which were not there before), it is impossible to get the entire set of real numbers.

You could end up with all kinds of interesting sets, though. One of them being the set of all rational numbers (any number which can be expressed as a fraction p/q of two integers, p and q). It sounds surprising, but in fact the set of rational numbers is only as large as the set of natural numbers. There is actually a way you can map every rational number to a unique natural number, even though if you look at them in the usual numerical ordering, there are infinitely many rationals between every pair of rationals!

I should write about this in another story; that should help clear up some confusion for people who aren't familiar with the concept of multiple infinities.

[ Parent ]

Infinity (2.00 / 1) (#315)
by gaweee on Mon May 26, 2003 at 12:37:12 PM EST

i'm not a math expert, but i seem to remember that : --> infinity*anything = infinity --> infinity/anything = infinity if u have 1 sphere of infinite mass, u can easily make 2 spheres of infinite mass because of the above rules. Then if that were the case, u wont even need rotation, just pure transposition is enough. since mass is related to density and vol. then all of the above holds true...... Heh, I know i think too simply. =D
he who runs away, lives to fight for another day
incorrect. (none / 0) (#316)
by warrax on Mon May 26, 2003 at 01:00:02 PM EST

infinity is not a number; there is no such thing as "infinity*anything" or "infinity/anything".

-- "Guns don't kill people. I kill people."
[ Parent ]
incorrect, again (none / 0) (#328)
by rickard on Mon May 26, 2003 at 06:15:55 PM EST

Actually, even though infinity is not a number in the ordinary sense, there most definitely is something like infinity*x and infinity/x, where x is a real or complex number. In complex analysis, it is often very convenient to work on the Riemann sphere where those expressions are valid and defined as the original poster seemed to remember. Also, x/infinity is of course defined as infinity. Try not to be so harsh next time.

[ Parent ]
stupid me (none / 0) (#329)
by rickard on Mon May 26, 2003 at 06:17:12 PM EST

What I meant was that x/infinity is defined as zero, not infinity, which is quite a lot more natural.

[ Parent ]
Nitpick (none / 0) (#342)
by anno1602 on Tue May 27, 2003 at 05:04:45 AM EST

To nitpick: There is no such thing as x/infinity = 0, x*infinity = infinity and so on. It is used in mathematics as a shorthand notation for saying:

Let x be an arbitrary constant number, y arbitrary. If y approaches infinity, then x/y approaches 0. Analogously, x*y approaches infinity.

But still, x*infinity is acceptable as shorthand as long as you know (and make clear that you know) that it *is* shorthand.
--
"Where you stand on an issue depends on where you sit." - Murphy
[ Parent ]

Nope (5.00 / 1) (#387)
by awgsilyari on Fri May 30, 2003 at 12:24:20 PM EST

This is how it's really stated:

    lim   x*n = sgn(x)*infinity
{n->infinity}

    lim   x/n = 0
{n->infinity}

Where sgn(x) is -1 if x is negative, and 1 if x is positive, and sgn(x)*infinity does not denote numeric multiplication -- it's a way to indicate which sign of infinity the result is.

The quantities x*infinity, x/infinity, etc. aren't directly defined, they are defined in terms of limits as some quantity increases without bound.


--------
Please direct SPAM to john@neuralnw.com
[ Parent ]

infinity/0 (none / 0) (#392)
by ThreadSafe on Sat May 31, 2003 at 11:25:50 PM EST

infinity/anything???

Anything???

Seriously though...

Make a clone of me. And fucking listen to it! - Faik
[ Parent ]

Banach-Tarski Paradox (5.00 / 2) (#337)
by lbianchi on Mon May 26, 2003 at 10:19:40 PM EST

A very good attempt to explain a rather subtle and abstract mathematical idea to the layman. Perhaps the only, minor, criticism is that the notion of "potentially infinitely complex" pieces or slices might have been expanded or explained a bit further. It's all there, but a bona fide layman may not see it. What I find depressing instead is that slew of idiotic comments and insults thrown at the author. Nobody was forced to read the article. Why bother commenting on something you don't care about?

This is easy to do in one cut (4.00 / 2) (#341)
by GoStone on Tue May 27, 2003 at 04:48:15 AM EST

Take the two subsets of rational* and irrational points. Each is infinitely dense and shaped like the original sphere.

There must be more to this paradox, surely?

*define a rational point as one whose x, y and z component is rational, though other similar definitions should also work.


Cut first, ask questions later

Not good enough (5.00 / 1) (#346)
by wurp on Tue May 27, 2003 at 10:25:54 AM EST

The set of all rational numbers is infinite on the order of the number of integers, not the number of reals.  The sphere composed of only points with rational coordinates would have measure (volume) 0.
---
Buy my stuff
[ Parent ]
thank you, but.... (5.00 / 1) (#348)
by GoStone on Tue May 27, 2003 at 10:44:02 AM EST

how about the set of points with at least one rational coordinate. The other coordinates could be irrational. Wouldn't this have the same order of infinity as the set of points in the sphere? After all a line can be mapped to an area or volume.

I'm sorry if these ideas seem obviously dumb. They don't to me, and I'm still trying to work out what is so difficult about the paradox.


Cut first, ask questions later
[ Parent ]

Oops, sorry! (5.00 / 1) (#350)
by wurp on Tue May 27, 2003 at 11:03:11 AM EST

I didn't read your post carefully enough; I thought you were only counting the points in which all the coordinates were rational.

That question doesn't seem dumb at all to me.  I would say, though, that if you do what you're talking about, you could find holes -- any of the points that you leave out would be holes.  I'm not familiar with the Banach-Tarski paradox, but for it to produce the same set of points, that means you can't identify a point that's in the starting sphere and not in one of the resulting spheres.

That said, your proposal looks to me as if it would produce two sets that each have volume equivalent to the volume of the first set.  But they wouldn't be spheres, since they're not the locus of all points within the radius from the center.
---
Buy my stuff
[ Parent ]

well (none / 0) (#351)
by GoStone on Tue May 27, 2003 at 11:28:24 AM EST

I think you read my original post right, I went and changed the rules slightly second time. But I can't agree with this:
...you can't identify a point that's in the starting sphere and not in one of the resulting spheres.
or this
But they wouldn't be spheres, since they're not the locus of all points within the radius from the center
I'm just reducing the cut from four pieces to two. Your objections would apply equally to the case with four pieces.

As for this:

you could find holes -- any of the points that you leave out would be holes.
I'm on shakier ground, but I'm fairly certain there wouldn't be any holes in the method I said.


Cut first, ask questions later
[ Parent ]
Holes (none / 0) (#354)
by wurp on Tue May 27, 2003 at 11:40:13 AM EST

Any point that has one or two rational coordinates and the other coordinates irrational would be a hole.  All the points count when you're talking about sets being equivalent.

The reason the Banach-Tarski can get past that is that they have multiple pieces and allow rotation and rearrangement of the pieces.  When you do that, it's possible that all of the holes from one piece are filled in by the other.
---
Buy my stuff
[ Parent ]

hmmm, (none / 0) (#356)
by GoStone on Tue May 27, 2003 at 12:09:56 PM EST

my two spheres would be infinitely dense all over. That is, any finite volume you might define, no matter how small, would have infinite density (as dense as the original sphere). In that sense there would be no holes, and they count as complete spheres.

My own suspicion is there is something we haven't been told! I mean this is not the first time I've heard of this paradox, and it is the best explanation I've yet seen (though I haven't looked far), but there still seems to be something missing.


Cut first, ask questions later
[ Parent ]

A single point missing is a hole (none / 0) (#357)
by wurp on Tue May 27, 2003 at 12:18:34 PM EST

It's not the "set of all points within 1 unit of the center" if the point (4,3,pi) is not in it.  I'm not saying that your resulting set has less volume than the original sphere, but I can't see how you could possibly say it's not missing points.  And if it's missing points, it's not a mathematical sphere.
---
Buy my stuff
[ Parent ]
Having read the link just given by rujith (none / 0) (#359)
by GoStone on Tue May 27, 2003 at 12:37:55 PM EST

I can see that you are correct. Not to put too fine a point on it (no pun). There was I thinking geometrically, when this is all about set theory. I tip my hat sir, and walk away almost, but not quite, as confused as I was originally.

(I mean this link)


Cut first, ask questions later
[ Parent ]

Infinite density (5.00 / 1) (#361)
by The Writer on Tue May 27, 2003 at 01:35:46 PM EST

Be careful when you say that something has "infinite density", and therefore must have no holes. The set of rational numbers is infinitely dense: between every two rationals there are an infinitude of other rationals. However, the set of rational numbers has "holes" where the irrationals would be. There are actually more "holes" than rationals (the set of all irrationals is infinitely larger than the set of all rationals), so actually the rationals are quite "sparse" when compared to the reals!

So it is not sufficient that you create a sphere which is as dense as the rationals; it is still "mostly holes" unless you are careful to fill them in with the appropriate irrationals. (Sounds impossible? That's why the choice sets in the Axiom of Choice are non-constructible.)

[ Parent ]

no, my two spheres are as dense as the original (5.00 / 1) (#374)
by GoStone on Wed May 28, 2003 at 05:16:49 AM EST

I understand what you say about the density of rational numbers, but the way I have formulated it my two spheres would both have the density of the real numbers, the same as the original sphere (at least I think so).

But apparently that is not enough, which is where my confusion arose. According to rujith's link a circle is not considered the same object as a circle with a single point missing, despite the fact that they both have the same density of points within any arbitrarily small finite section.

Personally I think this mixing of finite and infinite sets is anti-intuitive, and its not surprising you get weird results. To my mind a voluminous sphere with a single volumeless point missing is essentially the same object.

Not that I am criticising set theory (!) or indeed your article. It was very interesting. I wish there were more decent articles like it on K5. Shame about the boorish comments. It seems a lot of readers have a superiority complex.


Cut first, ask questions later
[ Parent ]

No dice (5.00 / 1) (#399)
by p3d0 on Mon Jun 02, 2003 at 08:39:09 AM EST

A sphere, by definition, contains all points within its radius. It's trivial to prove that each of your new "spheres" is missing certain points. For instance (taking radius=1), your "rational sphere" doesn't contain the point (0,0,sqrt(1/2)), so it is not a sphere.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
So how are the five pieces defined? (5.00 / 2) (#355)
by rujith on Tue May 27, 2003 at 12:09:07 PM EST

Great article, but I wanted to see how the five pieces are defined, even if it's just a sea of mathematics beyond my comprehension. A link to such a definition would be nice. Incidentally, here's a very good description of a gentler problem giving a nice flavour of how such pieces can be constructed: http://www.math.hmc.edu/funfacts/ffiles/30001.8.shtml. It describes how a circle with a single point removed can be equivalent to the original circle. - Rujith.

I think that is indeed the trick... (none / 0) (#379)
by ethereal on Thu May 29, 2003 at 08:48:08 AM EST

I'm not sure I agree with one of the central concepts here: how can a cloud of points distributed throughout the sphere be considered a "piece"? Shouldn't a "piece" be contiguous with all of its constituent parts/points? And why does one piece have to be the center point? (Is there even guaranteed to be a center point? You could argue that since the points are infinitesimal there will always be one, but I could argue that there's just a likely to be two infinitesimal points straddling the true center.)

I can see using "infinite density" to show how this works as a mathematical trick, but I don't think it's a very useful real-world paradox. You can have all sorts of paradoxes once you get infinity involved :)

--

Stand up for your right to not believe: Americans United for Separation of Church and State
[ Parent ]

Thanks (5.00 / 1) (#398)
by p3d0 on Mon Jun 02, 2003 at 08:35:26 AM EST

That is the most bindbending link I have read all week. The explanation is so easy to understand, yet so (apparently) impossible.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
This is exactly the kind of thing... (none / 0) (#364)
by Control Group on Tue May 27, 2003 at 01:52:20 PM EST

...which makes me wish I had taken classes past Discrete Math in college. You've given an excellent, accessible, elegant explanation - thank you - but I know just enough math to be unable to grasp how it's valid, and not enough math to make the last leap of understanding required to "get it."

Your analogy to a physical sphere is superb; it got me thinking along a track that makes the whole thing seem more plausible. My problem is that I can't get past the idea of "holes." I can't manage to see any way the two spheres aren't complete. I mean, if you take some random point in the original sphere (x,y,z) where x, y, and z are reals, and then translate that point into a second volume, I can't see how you get that point back.

I'm not putting this well. Say that one of the two spheres at the end exists in the same volume that the original sphere did (or perhaps that's implicit in the problem?). I don't see by what logic (x,y,z) exists in that sphere while its analogous (x',y',z') exists in the other. I can certainly accept that there are just as many points in each sphere as there were in the original, but I can't see how that point can be in both.

Hrm...unless it depends on the idea that the constituent points are uncountable, so a "neighbouring" point turns out to be the same point (I'm thinking along the lines of 0.999... = 1), so you can't "lose" a point in the translation...but if that's the case, then I don't see why you would need five pieces.

Speaking of which, that center point is the perfect example of my problem...how can one point become two? How can a point have any kind of infinite complexity?

*sigh*

I wish I was smarter.



***
"Oh, nothing. It just looks like a simple Kung-Fu Swedish Rastafarian Helldemon."

Answers? (none / 0) (#369)
by xee on Tue May 27, 2003 at 05:24:57 PM EST

The constituent points, like points on a cartesian plane, are of the form (x, y, z) where x and y and  z are real numbers.  That's what the funky R-cubed symbol means... the real numbers in three dimensions... the real numbers, cubed.  Your assumption that the points are uncountable is correct, as the continuum of real numbers, C, is an uncoutable set.

Remember, these kinds of proofs only assert that there exists some way of doing it... not that we'll ever know how.  Perhaps that branch of mathematics hasn't been invented (discovered?) yet.

As for the holes... well, you just have to accept that the sphere is infinitely dense, and can be divided infinitely many times.  The infinite complexity you mention exists in the depth of precision of the continuum, which is infinite.  Any real number can be divided in two, therefore, from one point, you can get two.  That's all this "paradox" asserts... big deal.

So, the logic is simple... if you have a sphere of radius 1 with a center at any point, r0, and then you cut each of its constituent points in half in all three dimensions, you'll have another sphere identical to the one at r0.  The key to understanding this is to keep in mind that any real number can be cut in half.  That's how one point can become two.

HTH.


Proud to be a member.
[ Parent ]

perhaps it's a bug (none / 0) (#366)
by AlfaWolph on Tue May 27, 2003 at 04:26:18 PM EST

Has anyone ever considered that our common system of base-10 arabic numeral mathematics is flawed and this amounts to an anomaly of sorts- a "bug" in the system if you will.

Or else... (none / 0) (#372)
by gniv on Tue May 27, 2003 at 11:29:27 PM EST

maybe you've seen Matrix Reloaded too many times.

[ Parent ]
Had a feeling.. (1.00 / 1) (#375)
by AlfaWolph on Wed May 28, 2003 at 10:25:51 AM EST

Yep. I knew some yokel would bring up the only thing he can remember having the word 'anomaly' in it. Some people's theories go beyond those proposed in The Matrix. As similar as it may sound to our dear architect's speech, this has nothing to do with The Matrix. Thank you for your troll. :)

[ Parent ]
You've hinted... (none / 0) (#391)
by ThreadSafe on Sat May 31, 2003 at 11:16:33 PM EST

at a very appropiate fact. Mathematics (anything dealing with integer values especially) is a construction of the human mind and not an inherent property of nature itself.

The whole system (of mathematics) depends on the assumption that various identical elements occur in nature. This is a fallacy and why a complete TOE (Theory of everything) or 'Theory of quantum gravitiy' if you will, will never succeed in being explain by mathematics (and consequently be comprehendible by a human consciousness).

Also note, that I don't beleive this is a reason to stop trying.

Make a clone of me. And fucking listen to it! - Faik
[ Parent ]

No (5.00 / 1) (#397)
by p3d0 on Mon Jun 02, 2003 at 08:33:39 AM EST

The Banach-Tarski Paradox has nothing whatsoever to do with base-10 arabic numerals.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]
More some aspects of mathematics in general (5.00 / 1) (#413)
by Craig Ringer on Sat Jun 07, 2003 at 09:03:23 AM EST

I can't agree that base10 or arabic numerals have anything to do with it (you can do the same maths in hex or using using dots in circles if you want ; it'll come out the same). However, some aspects of mathematics, while not /wrong/, may not really reflect reality. I sure as hell hope not.

Interestingly, all of the nasty ones I know about are based in set theory. Here are some interesting examples of why set theory might be ... less connected with reality ... than the rest of mathematics.



[ Parent ]
Um... (2.00 / 1) (#367)
by trhurler on Tue May 27, 2003 at 04:31:50 PM EST

Even if you don't take the Axiom of Choice as an axiom, given any reasonable definition of "sets," the result it predicts is provable. This is like arguing over whether you should define a spherical shell as the set of all points at a given distance from a given point or as the set of points coincident with a 180 degree rotation of a circle about an axis coincident with any chosen diameter. Either way, in the end, the same set of facts is reachable by further derivation - exactly the same set.

In fact, the biggest reason to reject the Axiom of Choice isn't because it creates "paradox" problems. The biggest reason to reject it is that given any reasonable set of axioms defining sets, this one is redundant.

--
'God dammit, your posts make me hard.' --LilDebbie

So... (5.00 / 1) (#380)
by hading on Thu May 29, 2003 at 10:50:15 AM EST

Why specifically do you feel that the Zermelo-Fraenkel formulation doesn't give a reasonable set of basic axioms for set theory?

[ Parent ]
Um, no (5.00 / 1) (#383)
by dcturner on Thu May 29, 2003 at 04:58:59 PM EST

That's why the A of C is controversial, you can't deduce it or any of its consequences from ZF set theory. Apparently.

Remove the opinion on spam to reply.


[ Parent ]
Anagram of Banach-Tarski (4.00 / 1) (#382)
by splitpeasoup on Thu May 29, 2003 at 04:58:32 PM EST

Q: What's an anagram of Banach-Tarski?
A: Banach-Tarski Banach-Tarski

This gem comes from my brother (who no doubt got it from somewhere else).

"Be the change you wish to see in the world." - Gandhi

Nice joke (none / 0) (#385)
by The Writer on Thu May 29, 2003 at 05:09:52 PM EST

Too bad somebody else already posted it.

[ Parent ]

oops (none / 0) (#386)
by relief on Fri May 30, 2003 at 03:31:20 AM EST

i posted a comment earlier that in retrospect, isn't relevant to this problem. so, the problem actually lies in the laziness of infinite uncertainty or something. ya well, just because you don't know where the hole will end up, doesn't mean it won't exist. i disagree with this whole article.

----------------------------
If you're afraid of eating chicken wings with my dick cheese as a condiment, you're a wuss.
maybe i didn't get it... (none / 0) (#388)
by Ginglith on Sat May 31, 2003 at 11:40:00 AM EST

but i dont' find it all that mind boggling. If you are using the idea of an infinitely dense circle, or a circle with an infinite number of points in it, I could very well take, let's say an infinitely long bridge and break it in half and we have two identical bridges. Its the same as your example of taking the set of Real numbers with it's infinite elements and splitting them up into Odd and Even. All 3 sets, Odd, Even and Real will be the same size because all three are infinite.

Yep you don't quite get it (none / 0) (#396)
by p3d0 on Sun Jun 01, 2003 at 10:56:04 PM EST

Your bridge analogy is apt, but I'm pretty sure your comment regarding reals and integers is wrong. There are more reals than integers. There is no way to pair up the reals one-to-one with the integers; that is, there is no bijection between the two sets.

There are different "levels" of infinity, and some infinities are bigger than others.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]

but then again... (none / 0) (#404)
by Ginglith on Mon Jun 02, 2003 at 02:03:05 PM EST

Hmm... how is the bridge analogy any different from the real/odd/integer scenario ? If you are imposing 'levels' of infinity you are then implying that the smaller 'levels' of infinity has a limit hence making it not infinite. (?) Just a thought...

[ Parent ]
no limit needed (none / 0) (#406)
by adiffer on Mon Jun 02, 2003 at 09:41:03 PM EST

You don't need to imply a limit on infinity to show that one type of infinity is bigger than another type.

Rational numbers are countably infinite.  That means you can make a map between them and the integers numbers and make it one-to-one.  Integers can be counted, hence the name.

Real numbers are uncountably infinite.  No one-to-one map exists between reals and  integers, so reals can't be counted.

A test to try to see if you understand would be to decide whether all the points in an infinite three dimensional space using real number lines along each of its three directions is countable, uncountable, or bigger than uncountably infinite.

(The fun little flip-side to this is that there are different sized differentials too.)
--BE The Alien!
[ Parent ]

not real.. but integers ! (none / 0) (#410)
by Ginglith on Wed Jun 04, 2003 at 09:49:31 AM EST

damn.. i see where my post has gone wrong.. I shouldn't have used the word 'real' as it implies that elements does not just consist of integers. I was wrong here. What I was trying to say is if we have a set consisting of all the integers (infinite) separate then into 2 sets, odd and even integers, shouldn't the set of odd, even integers be the same size as the set of all integers.. Now I'll go pop by and see what that new article on infinity is all about.. i might regret this post later.

[ Parent ]
The integers have gaps (none / 0) (#407)
by p3d0 on Tue Jun 03, 2003 at 09:55:41 AM EST

The reals are "continuous" in the sense that any two reals have another real between them. The same is not true for integers.

Thus, while integers are infinite in extent, reals are infinite both in extent and in density, and that means there are more of them.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]

I just remembered the proof (5.00 / 1) (#408)
by p3d0 on Tue Jun 03, 2003 at 11:05:24 AM EST

I just remembered the proof that there are more reals than integers. It's called a "diagonalization proof".

First, a definition: if we can pair things up with a subset of the natural numbers, then those things are said to be countable. For instance, I can pair up my fingers with the numbers 1 through 10, so my fingers are countable. Likewise, I can pair up the even natural numbers with the natural numbers (1->2, 2->4, 3->6, 4->8, ...), so the even natural numbers are countable.

Consider just the reals from 0 to 1. If they are countable, we should be able to pair them up with the natural numbers, so we can make a table containing each natural number beside its corresponding real number, written in decimal:

1 -> .183748761...
2 -> .187632476...
3 -> .430764872...
4 -> .298374918...
5 -> .713208723...
6 -> .612304598...

If the reals are countable, we can construct this table to contain every real number from 0 to 1.

Now consider the digits along the diagonal of this table. Suppose I create a new real number called x that contains those digits. In our example, it would have the value 0.180304...

Then, consider another number called y whose each digit equals 9-d where d is the corrsponding digit from x. That is, every digit is different from the corresponding digit from x. It would have the value 0.819695...

Note that y is unquestionably a real number. Therefore, it must appear in our table. However, it is different from every entry in the table in at least one digit, so it cannot be in the table. Hence, the table cannot contain every real from 0 to 1, and so the reals are uncountable.
--
Patrick Doyle
My comments do not reflect the opinions of my employer.
[ Parent ]

Wicked! (none / 0) (#390)
by ThreadSafe on Sat May 31, 2003 at 11:05:15 PM EST

Cheers for writing an informative, interesting and thought-providing article. Very good amongst most of the sludge that seems to be predominant on Kuro5hin these days!

Israel in the EU!!!!1 For fuck's sake; who gives a fucking fuck??!?!?1

Make a clone of me. And fucking listen to it! - Faik

a sphere question (none / 0) (#409)
by ireo on Wed Jun 04, 2003 at 07:55:04 AM EST

a 3-ball is defined by x^2+y^2+z^2<=1 (for closed ball), so interior of ball is by ....<1.
and the boundry of the ball will be ...=1, so is the
definition of the sphere.
am i right ? and the BT problem includes balls
or spheres?
 
A rock pile ceases to be a rock pile the moment a single man contemplates it, bearing within him the image of a cathedral. -- Antoine de Saint-Exupery
"Sphere" vs. "ball" (none / 0) (#411)
by The Writer on Wed Jun 04, 2003 at 03:09:06 PM EST

The BT paradox concerns solid spheres, or "balls" by your definition. I was debating whether to use "sphere" or "ball", but I settled on "sphere" since "ball" could mislead people into thinking about physical balls instead of mathematical balls. The original diary, which this article is based on, uses "ball" instead of "sphere", though.

[ Parent ]

re :"Sphere" vs. "ball" (5.00 / 1) (#412)
by ireo on Wed Jun 04, 2003 at 03:28:52 PM EST

sorry for this academic question. thanks
A rock pile ceases to be a rock pile the moment a single man contemplates it, bearing within him the image of a cathedral. -- Antoine de Saint-Exupery
[ Parent ]
Also works on a circle? (none / 1) (#416)
by dumky on Wed Aug 13, 2003 at 11:56:53 PM EST

You describe the ball example. Does this also work on a sphere or a circle?

If you pick a certain point (say 0.5,0.5,0.5) it can be in only one of the sets. What takes its place in the  second copy of the ball? I guess another point takes its place, but then the question becomes: what takes the place of this other point?

That's pretty much the point... (none / 0) (#417)
by falconheart on Mon Dec 22, 2003 at 06:37:44 PM EST

...as in the case of splitting the set of integers. Say I take set S={x|x is odd}. So what takes the place of the even numbers? For each even number, 2m, I map the number 2*(2m)-1; which belongs to S, to it. And what takes the place of 2*(2m)-1; well, it's 2*(2*(2m)-1)-1. We're kinda moving all the elements of S closer together to fill up the gaps.
So you say, "Well, what takes this numbers place, and so on, I can go on for ever!" But the point is, so can I.
So though there may actually be a "hole" in the system, by definition of the concept of infinity; "You can never find it, cos we can both go on forever.", at least not by this process of induction.
And since though there may be a hole in the new copy, I can never find it, no matter how much I may look, and so, for all practical and theoretical purposes, there is no hole. Kinda like taking the concept of "find" and "infinity" with a pinch of salt. Infinity has no place in real life.

For instance, if I tell you to pick a number, an integer, ANY integer, and if you manage to actually pick one, you've already negated the concept of infinity, since before you picked the number, the concept of infinity had claimed that the probability of you picking that particular number was 0. You've just done the impossible. Or have you?

[ Parent ]

Layman's Guide to the Banach-Tarski Paradox | 417 comments (354 topical, 63 editorial, 1 hidden)
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