Since the article doesn't explain how the sphere is split, we'd better describe it as a claim made by certain parties, much as the claim made by AGW supporters that their global climate models explain or predict anything. Since GCMs have never been explained in any encyclopedia article I've read, I assume they are too esoteric for us laymen to understand (or check). --Ed Poor Talk 19:15, 4 January 2010 (EST)
- Not at all. I can have a complete description of the Banach-Tarski paradox up by tonight which will be accessible to anybody who understands irrational numbers. I'll try and have it up by tonight. JacobB 19:30, 4 January 2010 (EST)
Purpose of Construction
Whatever the opinions of Banach or Tarski may have been on the validity of the AoC, we can relatively sure they did not develop the Banach-Tarski construction I describe in the article to comment on the AoC, because there was already a very weird AoC-based construction which had been well-known for 19 years prior to Banach and Tarski's publication of their result. The BTP may be used today to illustrate the consequences of the AoC more easily to a layman, but the original purpose of the construction was more involved with group orbits and (perhaps?) representation theory. I hope nobody minds if I make this minor change. JacobB 21:44, 5 January 2010 (EST)
- I think the introduction understates the motivation of the paradox. Wasn't it to illustrate how the AoC, if embraced, can lead to absurd conclusions?--Andy Schlafly 22:33, 5 January 2010 (EST)
No, it wasn't, but the Vitali set was. The Banach-Tarski paradox was originally developed to investigate group representations. Today, it is used to illustrate the "weirdness" of the AoC. Since the Vitali set WAS developed partly for the purpose you ascribe to the Banach Tarski paradox, I don't think we lose any of the strength of the original statement about the AoC by saying the Vitali paradox was developed to illustrate that weirdness.
If you like, I can have a full explanation of the Vitali set in that article up tonight. It deals more directly with why the AoC is necessary for the strange construction, and spends less time dealing with putting free groups on spheres and such things.
Summary: Although the Banach-Tarski paradox quickly and easily demonstrates the strangeness of the AoC, and I have no problem including that, that wasn't the motivation for its development. JacobB 22:44, 5 January 2010 (EST)
I'm not quite sure where the jargon is - all the terms I used, I define in layman terms, and to be fair, the mathy sections are labelled "appropriate for college undergraduates," which is in keeping with the idea of introducing community college level home course material. In other words, I'm not expecting a 8th grader to understand this.
If those users who find this jargony could indicate where the problem first begins, I can take care of it. I feel if a reader reads the article, they'll understand it perfectly. My advisor at my university taught this material to non-majors in a "great ideas of mathematics" style course. JacobB 17:29, 6 January 2010 (EST)
- I expect an 8th grader to understand this. Please state what part of it is too hard for an 8th grader to understand. --Ed Poor Talk 00:07, 8 January 2010 (EST)
- Well, for starters, most eighth graders are only just beginning their introduction to the idea of a variable. I think asking them to understand equivalence relations, let alone the axiom of choice or the elementary group theory, is a bit much.
- If you don't understand, tell what is confusing and I'll gladly fix it! If you do understand, I welcome your improvements to this article, to make it more accessible. JacobB 00:26, 8 January 2010 (EST)
- Can you explain equivalance relations and the axiom of choice, in terms of simpler concepts such as membership in a set, and so forth? Maybe you should work on basic concepts like set, empty set, union & intersection first. Many times users have signed up here and immediately started on university-level math articles, but I have rarely been satisfied with any of their explanations. I feel the flaw is a failure to communicate basic concepts. So, start on those, please. For example, there's no point in jumping to college algebra if we don't even have an article explaining how to solve equations in one or two variables.
- Spewing forth abstruse, incomprehensible symbols is not in line with our goals here. If you have a craving to speak the language of higher mathematics - but lack the willingness to make your points accessible to high school kids - then Wikipedia might be a more congenial place for you. --Ed Poor Talk 11:19, 8 January 2010 (EST)
Your edit is well-taken, Jacob, but I'm not so sure this impossible after all. "Logically possible but physically impossible" is not a claim I'm entirely comfortable with. This certainly provides food for thought!--Andy Schlafly 23:01, 6 June 2010 (EDT)
- The difficulty, of course, comes with the difference between "mathematical objects" and physical ones. I would say it's not just physically impossible with real objects, but logically impossible as well. The cuts required for a paradoxical decomposition are literally infinitely small and precise. Since real objects are made of indivisible particles which cannot be divided (quarks, electrons), and because their positions cannot be known with any great accuracy at that scale, the cuts could never be performed - it's not only a physical impossibility, but also a logical impossibility for any object made of a finite number of particles of non-zero size.
- With a "mathematical object," the case is of course different - these are more mental constructs than anything else, which we picture as being composed of infinitely many points, which are infinitely small. In order to perform a cut, we need only describe that cut. So in this situation, it IS logically possible. JacobBShout out! 23:09, 6 June 2010 (EDT)
- I do want to discuss this further, but need to cogitate about it more first. Your analysis above assumes that matter is composed of indivisible particles. I'm not sure that is completely known with certainty.
- Miracles are events that are logically possible but very rarely observed and unexplained by current understandings. But perhaps the obstacle is the understanding here, not the reality. And this "paradox" may provide a window that can be opened to a fuller understanding.--Andy Schlafly 23:44, 6 June 2010 (EDT)
Distinction between living and non-living matter
The hypothetical duplication works better with non-living matter, such as bread and dead fish, than with ever-moving living matter, for which the infinitesimal division would be impossible by virtue of motion.--Andy Schlafly 20:46, 8 June 2010 (EDT)