Talk:Balls and vase problem

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[edit] History

This article needs a History section detailing who first proposed it, historical arguments, etc. I have not yet found any good references on the web, but I remember seeing two or three good articles in the past. — Loadmaster 22:14, 6 November 2006 (UTC)

[edit] sci.math newsgroup

This problem has been discussed to death on sci.math. I've listed a few of the more recent (2005−2006) threads. — Loadmaster 22:14, 6 November 2006 (UTC)

[edit] Solution

The solution is 10-1+10-1+10-1+...

By straightforward arrangement, it is also 1-1+1-1+1-1+...

Therefore, according to Bolzano's Paradoxien des Unendlichen, the serie can be prooved to be any number and the problem has no mathematical solution.

AlainD 08:57, 13 January 2007 (UTC)

On a sidenote, noon is expected to arrive at 2^(-(n-1)). According to the problem description, n is countably infinite. Thus, n->infinite. Which means 2^(-(n-1))->0. That is also one point that illustrates that the problem's initial assumptions were problematic, a fact that is intuitively confirmed. (essentially - and if I'm mistaken please post - "how many steps of 9 does it take to count to infinity ?") —Preceding unsigned comment added by 87.202.27.214 (talk) 02:44, 7 October 2007 (UTC)

[edit] Article title

Would this article not be more accurated titled Ross-Littlewood paradox? Also, Supertask#Ross-Littlewood_paradox contains more good information than this article; some content really should be moved to here, leaving a shorter summary in the Supertask article. ~ Booya Bazooka 04:01, 25 January 2007 (UTC)

That's probably the best approach, to merge this article with Supertask#Ross-Littlewood_paradox into a new Ross-Littlewood paradox aticle. Then the contents of the section in Supertask can be condensed, being replaces with a "Main article" link to the new article. — Loadmaster 22:01, 15 February 2007 (UTC)

[edit] References in BJPS

The British Journal for the Philosophy of Science has two relevant publications regarding this topic:

  • "On Some Paradoxes of the Infinite". Victor Allis; Teunis Koetsier. Vol. 42, No. 2. (Jun., 1991), pp. 187-194.
  • "Ross' Paradox Is an Impossible Super-Task". Jean Paul Van Bendegem. Vol. 45, No. 2. (Jun., 1994), pp. 743-748.

Electronic versions are available on JSTOR if you have access to it. ~ Booya Bazooka 04:28, 25 January 2007 (UTC)

I went ahead and added those two references to the "References" section of the article. They need to be wikified into ref form, though. — Loadmaster 17:19, 22 February 2007 (UTC)

[edit] The Lamp Paradox

Shouldn't there be a mentioning of the Lamp Paradox? It is basicily the same as the Vase Paradox, it goes at follows: "Let there be a Lamp which can have two states: "on" or "off". At t=1 the lamp is off, t=1/2 the lamp switches on, t=1/4 the lamp switches off and so on. The question posed is: what will the state of the lamp be at t=0?"

One of the suggested answers is that the lamp is neither on nor off, because it never reaches t=0. Maest 20:19, 4 March 2007 (UTC)

Well, the two problems are similar, but they are not the same. The first asks for the final sum of balls based on adding and removing them from the vase, while the second asks what the "final" state of the lamp will be. The first question can be answered assuming that the time t=0 can be defined well enough. The second cannot be answered because even if time t=0 is defined, the answer relies on knowing the "last" step prior to t=0, when the lamp was switched on or off for the "last" time. The first problem needs no "last" step to be answered. It would be nice to have a "Lamp Paradox" article that could be added to the "See also" section, though. — Loadmaster 00:31, 6 March 2007 (UTC)
I found the article, named Thomson's lamp, for the paradox you describe. I've added it to the "See also" section appropriately. And I added a redirect for Lamp paradox as well. — Loadmaster

[edit] Pearl of Wisdom

A variation of the Balls and Vase problem is the Pearl of Wisdom. An example of the tale can be found at www.246.dk/pearl.html. — Loadmaster 22:03, 13 August 2007 (UTC)

[edit] My 2 cents

I think this problem simply shows that you have to be careful about your terms - in particular, when talking about limits, you have to define your topology first.

The problem can be phrased like this: We have a function f from [0,1] to some state space of the vase, let's call it X. We have defined f on a sequence converging to 1 and would like to know the continuous continuation.

If we let X be the one-point compactification of the real numbers and f(1-2^-n)=9n, then the limit is infinity.

On the other hand, if we let X be 2^\mathbb{N} (each component representing the presence or absence of an individual marble) with the product topology or in other words the topology of pointwise convergence, and f(1)=(1,1,...,1,0,...) (10 ones, then zeroes) and f(2)=(0,1,1,...,1,0,...) (0, then 19 ones, then zeroes) and so on, then the limit is the zero sequence, since every component pointwisely converges to zero.

You see, it's just a matter of proper definitions. If there is a notable source for this argument, I strongly suggest adding it to the article. Functor salad 04:10, 16 October 2007 (UTC)

[edit] Benardete's Paradox

Another variation is Benardete's Paradox, said to be an extension of Zeno's Paradox, which is described at thinkquest.org. Briefly, the god Zeus decides to kill Prometheus by issuing an order to an infinitude of his demons, ordering the first demon to kill Prometheus if he is not dead by 2:00, the second to kill him if he is not dead at 1:30, the third to kill him if he is not dead at 1:15, etc., each demon killing Prometheus in half the time interval of the subsequent demon. The strangeness comes in the realization that Prometheus is indeed dead by 2:00, but yet no specific demon killed him. — Loadmaster (talk) 04:11, 30 March 2008 (UTC)

[edit] Fake question

Just like that infamous lamp, this question is a mathematical/philosophical/logical question masquerading as a physical one. Obviously when you're talking infinities, "balls" and "vases" are just placeholder concepts: the number of physical operations that can be performed in a finite time is finite. Casting this problem in physical terms is not valid. A more honest question is "what is the limit of the infinite sum "10 - 1 + 10 -1 ..."; this question can be mathematically discussed. A question of balls and vases falls at the basic hurdle of making physical sense. --Slashme (talk) 15:33, 27 May 2008 (UTC)

Since the best evidence so far is that the Universe is composed of discrete matter-energy/time-space quantum "units", any mathematical problem dealing with continuous functions, real numbers, or infinities is bound to fail to make total sense as physical reality. — Loadmaster (talk) 00:01, 28 May 2008 (UTC)
Actually, some proponents of hypercomputation are hoping that at some point in the future there will be actual, physical computers performing infinitely many operations (in sequence) in a finite amount of time: a physical Zeno machine. —Preceding unsigned comment added by 128.113.89.96 (talk) 15:21, 28 May 2008 (UTC)
Well, when that time comes about, it'll be interesting to see what result comes out. From Wikiquote:
We all know Linux is great…it does infinite loops in 5 seconds.
-- Linus Torvalds about the superiority of Linux on the Amsterdam Linux Symposium

--Slashme (talk) 05:21, 30 May 2008 (UTC)