Talk:Zeno's paradoxes

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[edit] Old,unsectioned comments

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These paradoxes can also be explained by quantum physics, since certain particles can only occupy discrete positions and can "move instantaneously" from one point to another without crossing the space between. (this is known as "tunneling" and is related to the "probability wave" nature of particles at the subatomic level.)

Sorry, I don't think this does anything to resolve the paradox. In quantum mechanics, position is a continuous variable. If the person who added this thinks it is correct, please explain further.CYD

You are the quantum physics guy, not me, and I didn't even write the stuff removed but:

Zeno's paradox is based on the given that to get from point a to point b, you have to pass through all points in between. Since position is continuous, thats impossible. If quantum physics says (and I have no idea if it does truly or not) that a particle can travel from point a to b without traversing the points in between, then a particle is not going to be bound by Zeno's paradox. That said, someone would have to make the case that ALL motion by particles is achieved by tunneling. Is this the case? Again, I'm not a physics person so I don't know, but I suspect thats never been proven or demonstrated. Again, since position is continuous, this could never be observed, so it would have to be shown theoretically at best.

--- "That is, quantum mechanics may prevent us from making infinitely precise measurements, but if time and space are continuous, the paradoxes still apply." Remember that quantum physics is a description of physical reality. The discreteness of energy is not a mathematical tool but the way energy comes. Similarly if time and space are discrete in a fully-proofed theory then they probably are in reality. There is no point talking about what is impossible fo the particle (infinite division of space) because it won't be possible for anything else. This will effectively make the paradoxes irrelevant.

Currently quantum theory does not use discrete time and space, though this may be because we haven't hit the experimental limits yet. There is plenty of theoretical evidence that distances below the Planck length and times shorter than the Planck time are meaningless.

Copies of the newly released papers by Peter Lynds talking about this subject can be found at the following locations: at http://cdsweb.cern.ch/search.py?recid=624701 and "Zeno's Paradoxes: A Timely Solution" is available at http://philsci-archive.pitt.edu/archive/00001197/

A brusque dismissal of Lynds's approach by me can be found at http://sl4.org/archive/0308/7012.html -- mitch, not yet a user 24 Jan 2004

Brusque is right. Going by your dismissal, you might as well have just said "I don't like it because I don't like it." Better still, "I don't like because I don't understand it".

Why the page move (from Zeno's paradoxes)? I mean, there's more than one of them... --Camembert

I was thinking of wikipedia:naming conventions (pluralisation), but re-reading that it's not as simple as I had thought! Move it back if you like. Martin 20:32, 23 Sep 2003 (UTC)
I do like, and I shall move :) --Camembert


[edit] Re: Peter Lynds

To the anonymous editor (User:81.80.88.113 etc.):

Please read Wikipedia:What Wikipedia is not and Wikipedia:No original research. Then please list on this page, which authoritative texts and/or prominent scientist/philosphers lists Peter Lynds as having solved Zeno's paradoxes. You might also want to mention, why they did not think it already was solved by calculus.

When you have done so, read Wikipedia:Neutral point of view. Then you can add to the article that these people believe that the paradoxes needed solving, and how Peter Lynds managed to do it.

Rasmus Faber 13:16, 28 Jan 2004 (UTC)

[edit] A few I quickly found

"Author's work resembles Einstein's 1905 special theory of relativity", said a referee of the paper, while Andrei Khrennikov, Prof. of Applied Mathematics at Växjö University in Sweden and Director of ICMM, said, "I find this paper very interesting and important to clarify some fundamental aspects of classical and quantum physical formalisms. I think that the author of the paper did a very important investigation of the role of continuity of time in the standard physical models of dynamical processes." He then invited Lynds to take part in an international conference on the foundations of quantum theory in Sweden.

Another impressed with the work is Princeton physics great, and collaborator of both Albert Einstein and Richard Feynman, John Wheeler, who said he admired Lynds' "boldness", while noting that it had often been individuals Lynds' age that "had pushed the frontiers of physics forward in the past."

http://www.newscientist.com/opinion/opletters.jsp?id=ns24249 http://ciencia.astroseti.org/astrofisica/entrelynds.php http://www.space.com/scienceastronomy/time_theory_030806.html http://gauntlet.ucalgary.ca/story/6121 http://www.thescotsman.co.uk/international.cfm?id=827792003 http://www.dagbladet.no/kunnskap/2003/07/31/374849.html http://www.physics4u.gr/articles/2003/lynds.html http://perso.wanadoo.fr/marxiens/sciences/lynds.htm http://www2.uol.com.br/cienciahoje/chdia/n935.htm

Page entry already indicates what is wrong with the calculus solution (assumes determined position at each instant (and the existance of instants), so doesn't actually solve paradoxes and show how motion is possible) - it's just a mathematical trick to get rid of the infinity.

There are multitudes of scientists and philosophers who don't think the calculus approach provides a solution (obviously more so since Lynds work). Just have a look around the web, read a book on the paradoxes, or read the work of someone like Bertrand Russell.

Repeatedly quoting Khrennikov out of context, do not exactly increase your credibility. He also said: "It's interesting but it's not great." Wheeler might have had the credibility needed, but he has not claimed to accept the conclusions of the paper, only that he "admired Lynds' boldness". (Note, however, Decumanus' objections on Talk:Peter Lynds).
If you feel that the article incorrectly claims that calculus solves the paradoxes, you might want to add to the article why Russell questioned this solution. Something about Grünbaum and McLaughlin would probably also be appropriate. But do not add Peter Lynds.
Rasmus Faber 08:39, 29 Jan 2004 (UTC)

[edit] You're mad

You're absolutely mad Rasmus. How could that khrennikov quote be out of context?! Lynds is right. Even if you disagree though, it should be included. It obviously deserves to be. What's your problem? personal attacks snipped

Politeness, Mr. Lynds, is always in order. — No-One Jones (talk) 13:32, 29 Jan 2004 (UTC)

I apologize. It was not the Khrennikov quote that was out of context. It was the quote from the referee. You forgot to mention that the comparison to Einstein's paper was not about any particular brilliance or importance, but to defend the validity of the circular reasoning.
Did you read Wikipedia:No original research? It explains Wikipedia's policy as to which theories deserves inclusion. Now please let me know which prominent people thinks Peter Lynds' research is important in the context of discussing Zeno's paradoxes. I don't want a listing of which popular science magazines have had articles about him. Nor do I want a listing of which scientists have similar theories. I want names and quotes from a few prominent scientists who say, that Peter Lynds' theory is an important new solution to Zeno's paradoxes.
Rasmus Faber 13:53, 29 Jan 2004 (UTC)

What more do you want Rasmus? Lynds to firstly get a Nobel Prize? Khrennikof and the other referee made those comments about Lynds' paper.....it contains his solution to the paradoxes and much of the rest of the content is based upon the same reasoning and argument. The journal editor obviously also thought it important. Throw in general support for his work by the likes of Wheeler and Davies (and I'm sure many more), and his solution definately qualifies as "significant scientific minority".

As a side note, the referee who made the Einstein comparison obviously meant that the paper's arguments were still valid even if possibly circular.

[edit] Tortoise

This isnt a paradox, its simple. First you have to define the size of the tortoise. Next you have to decide when the distance between the tortoise and achilles is less than that size. Bensaccount 00:40, 14 Mar 2004 (UTC)

Understanding the common sense portion of the paradox is not hard. The point is that mathematically he could never pass the turtle. Brynstick 18:14, 24 January 2007 (UTC)

[edit] Dangerous minds

I found the associated work dangerously confident in its statements. As is correctly stated, Zeno based his paradoxes on the work of Parmenides. Reading 'Parmenides' by Plato, one is reminded that Zeno's examples were to show that Parmenides logic was true because if it was false, the universe as we know it is even crazier than if Parmenides is right. One should always bear in mind that mathematics is based on axioms and is subject to Godel's Incompleteness Theorem and is fraught with internal paradoxes and ambiguities, many of which are seriously close to the areas related to Zeno's paradoxes. My edit was minor, only intended to moderate the view that Zeno has been dealt with. Any quantum theoretical treatment (or use of infinities, limits etc) that purports to resolve Zeno's paradoxes, is founded in Parmenides 'World of Seeming' (his 'Way of falsehood') and so is no closer to resolving the foundations of the paradox at all. Lot's of people seem to miss this point. Zeno was not setting a mathematical problem, but a problem for the universe as we think it to be (eg quantum models and Hilbert's formalism). Centroyd 31 May 2004

  • HUH?..."not setting a mathematical problem"...a paradox IS, by definition, a mathematical problem. BTW...there is no such thing as a paradox, only an unsolved problem.

[edit] This page needs a big edit

Rereading this page makes me nauseous. It is so full of people being certain about things that that are deeply flawed. Consider the Lynd's 'solution' that is put up as a solution to the paradox. I do not suggest that Lynd's is not right in limiting infinitessimals, but this is not a solution to the paradox - quite the opposite. If what I read on the page is representative of Lynd, then he is saying there are no infinitessimals, therefore there is no paradox. But Zeno would say 'So what? That is not my point.' As previously commented, a solution to Zeno needs to consider Parmenides in conjunction. In the meantime I think this article needs to be more balanced. Centroyd. 1st July 2004.

[edit] "The rock thrown towards a tree"? What is the source for this??

I've never heard of this paradox before. What is the source for it? I'm very familar with Aristotle's Physics and I don't find it there. Nor do I find it in Simplicius's commentary. I'd like to replace it with the "Dichotomy" paradox given by Aristotle. Any objections?

--

Yes, I agree to replacing it - but Dichotomy is not very descriptive either, though it is the traditional heading. This is the sum of an infinite series of infinitesimals & the easiest to present arguments against - unless you are the subject walking towards the tree (or wall) & try to take just the first step.

The entire article needs to be laid out better. The arguments trying to overcome the paradoxes are presented before all the paradoxes are even presented -- and are not even clearly distinguished from them. --JimWae 05:17, 2004 Nov 17 (UTC)

[edit] Physical explanations afterthought

Just thinking after reading the section on physical explanations. "Lynds asserts that the correct resolution of the paradox lies in the realisation of the absence of an instant in time underlying a body's motion, and that regardless of how small the time interval, it is still always moving and its position constantly changing, so can never be determined at a time." First off, I think that sentence needs an edit. I'm not sure how to do it though. Also, isnt that precisely the fundamental calculus assumption? That regardless of how small the limit of time tends to, the velocity need not tend to zero, and can have a finite value. Thus even at a "still" moment, the arrow's position is changing at that finite rate. Or am I missing something? Ethan Hunt 00:40, 2 Sep 2004 (UTC)

[edit] Rival article

Someone just added Xeno's paradox. Might want to compare/merge. Ortolan88 18:33, 14 Nov 2004 (UTC)

Now merged. Author of Xeno article can retrieve, merge content if they wish. -- Decumanus 18:37, 2004 Nov 14 (UTC)

[edit] The practical aspects

Surely the retort to the tortoise paradox is - Achilles' foot has a specific size, and there is, therefore, a physical minimum distance which he can move?

The arrow exists in actually existing time - and so will move.

I know these are theoretical exercises, but the practical reasons why they are not paradoxes in the real world could be included.

Jackiespeel 22:05, 3 August 2005 (UTC)

The point to the paradox is putting practicallity aside. We all know that motion is possible or that anyone can run past a turtle, the point is the mathematics. Brynstick 18:16, 24 January 2007 (UTC)

[edit] Alternative Explanation of the Tortoise Paradox

The explanation with the geometric series is not sufficient. It proposes a solution but fails in explaining why the original proposal was paradoxical. Therefore doesn't solve the paradox. To be an acceptable explanation a solution has to show a flaw in logic or assumptions of the original proposal.

The paradox can be explained in concepts used in Mathematical Analysis, namely of metric spaces. The events described are equvalent in trying to analyse an open segment against a colosed one. For example [0,1) against [0,1]. The description of events in the paradox is restricted to the segment [0,1) but the analysis extends to [0,1] namely to the last point (1) in the segment. Asking when Achilles reaches the tortoise is equivalent to asking "What is the larges number in the (open) segment [0,1) ?" or "What is the largest number that is smaller than 1?" Of course there is no such number. The paradox assumes that in order Achilles to pass the turtoise a number (or moment) like this has to exist but space cannot neccessarily be devided into finite elements. The two segments [0,1) and [0,1] are not identical though the difference is infinitessimal. An invalid assumption and therein lies the paradox.

Martin, Sydney, Australia


I think this whole article is ridiculous. It's like having an article about how the sun produces its energy and prominently describing calculations that show the sun would soon be gone if it got all its energy from coal burning, then listing nuclear fusion under "proposed explanations". Ken Arromdee 17:11, 2 September 2005 (UTC)

RE:

It is not ridiculous. This paradox is important in reconciling human cognition with the real world which is the aim of any natural science. Scientist build models of the real world and try to test them empiricly. Most engineering invetions are based on inventions in physics, most inventions in physics are based on inventions in mathemathics and many mathematical inventions arose from philosophy. See Russel's Paradox and Mathematical Logic/Axiomatization.

It is ridiculous. This is just called a "paradox" for historical reasons, but "Zeno's fallacies" would be a more accurate title. It just shows how a process as simple as motion at constant speed can be obfuscated by resorting to misconceptions about infinity. Itub 16:55, 3 February 2006 (UTC)
I agree. Zeno's paradoxes represent neither human cognition nor the real world as seen through any natural science. You are never going to see any practical application of it. It is more like the difference between understanding something, and venerating the confusion.Algr 18:56, 3 February 2006 (UTC)
It's not just the title. The problem is that the article treats the actual solution to the paradoxes as just one among many solutions that may or may not be right, then goes on about how mathematicians "thought" they had solved the problems and "It would be incorrect to say that a rigorous formulation of the calculus... has resolved forever all problems involving infinities, including Zeno's". We don't have articles about the sun which imply that nuclear fusion hasn't completely solved the problem of where the sun's energy is from, and we don't have articles about evolution which say that scientists "thought" that men evolved from apes but evolution doesn't completely explain how human beings got here. The idea that Zeno's Paradoxes haven't been solved is an extreme minority view and should not be treated as anything more than that. This article heavily violates No Undue Weight (link isn't very good--you have to scroll down yourself) Ken Arromdee 18:56, 24 February 2006 (UTC)
And this is now a "good" article. Sheesh. Ken Arromdee 20:14, 2 August 2006 (UTC)

[edit] Pile of Sand

I'd like to see some treatment of the pile of sand paradox. i.e. One grain of sand is not a pile of sand, and if you have something that's not a pile of sand, adding one grain of sand isn't going to make it a pile. Therefore, you can never have a pile of sand. (Saying there is such a thing as a pile implies that there is a precise point at which not-a-pile becomes a pile, and most people would regard saying that 1253 grains of sand is not a pile, but 1254 grains *is*, as absurd.)

I'm not adding it to the text myself, as I'm not sure if it considered equivalent to one of the other three paradoxes, but there should at least be some mention of the paradox in that form somewhere on Wikipedia, as I have encountered that formulation a number of times. (And it is interesting to think about, at least from the perspective of what we mean when we say "pile".) -- 17:21, 13 September 2005 (UTC)

[edit] The Dichotomy Paradox ("You cannot even start.")

I think the Dichotomy Paradox simply shows that time and space are not infinitely divisible. Rather than showing that motion is an illusion, it shows that continuous motion is an illusion. Displacement over time is actually composed of discrete quantum leaps of finite size, similar to how a mouse cursor moves across a computer screen in multiples of single pixels. The concept of quantum motion has been essentially validated by quantum physics via the Planck length and Planck time.

The argument that each successive division of distance requires half the time to traverse, doesn't explain how the runner starts moving in the first place. If he's not in motion, time is irrelevant. For any given displacement, there remains half that distance to be crossed first, ad infinitum, unless there is a minimum distance that can't be further divided. Owen Ward 23:35:00 UTC, Wednesday, September 17, 2005

  • Relevant to this is the issue of whether time and space are entities at all or rather (unavoidable) intellectual constructs (like number --JimWae 01:31, 18 September 2005 (UTC)

[edit] Order & explanation

I moved the 'References' before the 'See also' section, as this is the specified order in WP policy. I also added an explanation sentence to make it clearer what the paradox actually is. Uriah923 07:43, 25 September 2005 (UTC)

[edit] A Simple Solution to the Arrow Paradox

When studying physics, a student in my high school asked the professor about the arrow paradox. The professor couldn't find any logical way to defy the description of the paradox, and was so upset he refused to give any lesson for the rest of the period. The next day, he brought a bow and arrow into the room, and show an arrow at the wall. Then he said "Who here can prove that the arrow didn't move?"

RE:

This is not a solution to the paradox. The arrow moved, no question about that, but this doesn't explain why the reasoning mentioned in the paradox was wrong.

[edit] Bizarre Definition of Infinity

I can't be the first person to have thought of this. It seems to me that most of Zeno's paradoxes are based on the same fallacy:

- Any task is infinite if it can be divided an infinite number of times.

The example of Homer catching the stationary bus makes this the most clear. Homer is performing what most people would consider to be a single task - move 20 feet to the bus. Zeno insists that this is impossible, because first Homer must move half the distance to the bus, then half the remaining distance, and so on infinitely. But nothing in Homer's action reflects this pattern - he is moving a finite distance at a constant speed. Homer is NOT performing an infinite number of tasks, it is Zeno who is performing an infinite number of measurements. So why should Homer be constrained by Zeno attempting to redefine "20 feet" as an infinite number by measuring it an infinite number of times? This is a bizarre definition of infinity.

Because Zeno's measuring system is inappropriate to the event being measured, the event cancels out Zeno's infinity by providing an inverse one: As the remaining divisions approach the infinite, Homer crosses them at approaching infinite speed. Thus the calculus solution to Zeno.

I've found some references that sound sort of like this, but they aren't really close enough. I'll keep looking. Algr 09:14, 15 January 2006 (UTC)

This is very amusing. The article contains a huge mish mash of contradictory ideas, unnecessary mathematics, and appeals to obscure physical theories. Anyone reading the article would be thoroughly confused about the real status of Zeno's "paradoxes". But then you find a clear description of the resolution to the "paradoxes" in the discussion page. The section "Problem with the calculus based solution" gets to the nub of the problem and the above resolves it, so this section should be rewritten, retitled and promoted to the top of the proposed solutions section. In fact the rest of the proposed solutions should probably be relegated to the discussion page. Pseudospin 12:44, 13 May 2006 (UTC)

Ha ha! The reason I haven't done that is that I can't find a source for it. For all I know I may have invented that whole solution. Algr 15:29, 13 May 2006 (UTC)

[edit] Missing some basics

Basic things seem mising from this article.

What happened to the Stadium?

[The Stadium: Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time.]

What happened to the larger paradox?

"We arrive at Zeno's paradox only when these arguments against infinite divisibility are combined with the complementary set of arguments (The Arrow and The Stadium) which show that a world consisting of finite indivisible entities is also logically impossible, thereby presenting us with the conclusion that physical reality can be neither continuous nor discontinuous."

See, for example Zeno and the Paradox of Motion in Reflections in Relativity at http://www.mathpages.com/rr/rrtoc.htm

Actually, I'm sure I heard somewhere that the four paradoxes were designed to cover the four possible combinations of {discrete time, continuous time} x {discrete space, continuous space} - Stadium is (DT,DS), Arrow is (DT,CS), Dichotomy is (CT,DS) (I think) and Achilles & Tortoise is (CT,CS). Then the above quote is correct in that he has refuted (to the extent of mathematics at the time) all possible interpretations of space and time. Confusing Manifestation 02:39, 24 February 2006 (UTC)

[edit] Missing some basics

[edit] More paradoxes

What about his claim that space can't consist of zero-sized points, since if you keep adding zero it still equals zero? The reply to this is that since an infinite amount of zeroes is uncountably infinite (as opposed to countably infinite, as in infinite sets), the sum of uncountably many zeroes is undefined (it's uncountabley infinite). 70.111.251.203 20:28, 18 February 2006 (UTC)


[edit] Question

The paradoxes all seem to stem from the same idea: That at any instant in time, nothing is happening. At an exact moment, Achilles is not in motion but at the exact spot where the turtle used to be. Then couldn't you call Zeno's basic paradox "Movement doesn't exist unless you acknowledge the passage of time"? Invisible Queen 14:53, 12 March 2006 (UTC)

[edit] Modeling movement and space and philosophy of physics

I think the article should enphasize more that Zeno's paradoxes can be thought to mean a more deep question of whether our mathematical models of space (for instance we usually assume ZFC set theory, and that space is locally describable by a R3 space) and motion(mechanics) are mere useful descriptions to predict fenomena or really have the reality we usually give to them (meaning that reality is described by some exact mathematical model). Or we could also question if he is worried about the contrast between theorical models (such as geometry and arithmetics where we can divide indefinitely and talk about infinities) and our common sense comprehension of reality ("Achilles will achieve the turtle and then...") where things are finite. Or even more, he could be questioning what kind of assumptions are reasonable when talking about space and time.

[edit] 10 April 2006

I have removed edits by an anonymous IP. Early edits were questionable original research; I reverted when the author signed the article. Feel free to clean up the formatting and resubmit, provided that a source can be found. Isopropyl 23:24, 10 April 2006 (UTC)

[edit] Proof by notation

Application of mathematical notation doesn't solve a problem. I'm tempted to remove the "Solution using calculus notation" section entirely. Fredrik Johansson 14:40, 3 May 2006 (UTC)

Doesn't the proof of calculus formulas just ignore infinitesimals? Rather than remove, to prevent re-insertions, mention of its shortcomings --JimWae 15:12, 3 May 2006 (UTC)

I think it's not a solution but still relevant to the article since it's a very widespread misconception. A solution to any logic paradox must show either that
1. one of the assumptions is wrong or
2. something is wrong with the reasoning.

Only discarding the assumption of infinitely divisible time/space in the real world satisfies the above.

The assumption that is wrong (to any mathematician) is that an infinite sum cannot be finite. Itub 16:29, 17 July 2006 (UTC)

That is only the assumption of the 'weak' paradox (see the section on 'issues with the calculus-based solution). The 'strong' paradox merely points out that one cannot reach the end of something that has no end; calculus does absolutely nothing to resolve the stronger paradox.

[edit] Paradox in popular culture/Dilbert

I've decided to be bold and I've added the Dilbert reference. However, (1) I am not absolutely certain that it is relevant and (2) is it copyvio? The Dilbert entry already contains an image of the strip so I think I am legal. (cubic[*]star(Talk(Email))) 20:31, 3 May 2006 (UTC)

Silly me, I edited this earlier today but forgot to check the talk page. I think it's quite appropriate, and such a short quotation is definitely fair use. —Keenan Pepper 02:42, 4 May 2006 (UTC)

[edit] Incomplete sentence

I was going to make a note about an imcomplete sentence, but I see someone has beat me to it.

"Classical mechanics asserts that things such as rate of change of acceleration of at an instant."

I think that's supposed to be "Classical mechanics asserts that things such as rate of change of acceleration can be defined at an instant," because it means nothing in its current form. -- 70.81.118.123 02:52, 7 May 2006 (UTC)

[edit] Remove?

I thought the two paragraphs recently added to the "Problem with calculus-based solutions" were inappropriate:

[On the other hand, it is conceivable is that there is no reason why an infinite number of events cannot be "finished."]

There *is* a reason, and it was stated several times right above this paragraph: to finish a sequence that has no finish is a logical contradiction.

["Finishing" a set of things means to do all of those things in a finite amount of time. If an infinite number of things can be done in a finite amount of time, then those things are all "finished." It could likewise be argued that this is exactly what the calculus solution demonstrates.]

Right, that is the calculus-based solution, but this section shows a problem with exactly that solution: that calculus merely points out that the sum of an infinite number of terms can be finite, but that it does not explain how one can ever be done with going through an infinite series. The section also explicitly includes a discussion of how considerations of time do not help. If there is a genuine response to these problems, it can be posted, but the paragraph as is merely repeats the calculus-based solution. I think it should be removed.

[In this point of view, the number of "acts" involved in anything is merely a matter of human convention and labeling. One could alternately consider the whole sequence to be one "act." Calculus demonstrates that the process takes the same amount of time and is "finished" in exactly the same way with either manner of labeling. This could be taken as evidence that the labeling of acts is arbitrary and has nothing to do with the underlying physical process being described.]

This comment seems to be not any different from the "Conceptual approaches" section, so it should be placed there, or removed.

128.113.89.43 17:48, 11 May 2006 (UTC)

[edit] Quality of external links

I don't see the point of including a random set of links to various quality sites mentioning Zeno's paradoxes. I've read through them all and the following three seem particularly poor

The first two completely miss the point of the paradoxes, and the third is written in a very pompous manner and takes nine pages to say that we cannot in practice measure instants in space and time.

Why link these people's musings ahead of the 293000 other hits google gives for "Zeno's paradox"? -- Pseudospin 09:40, 14 May 2006 (UTC)

[edit] "Questions raised" Section removed here

This entire section is filled with editorial comment (POV) AND interrupts exposition of the paradoxes. If any of it belongs, it belongs towards END of entire article ---

[edit] Questions raised

Zeno appears to be presenting situations which conflict with our intuition. We are invited to consider the nature of motion. Can motion be defined for an instant of time? Zeno indicates that for a point in time there is no motion (arrow paradox). In the race he notes changes in position but ignores their relative magnitudes over intervals of time. So there appears to be no qualitative change in the circumstances of the race. One must also ask if the contestants are being treated equitably. To capture the situation better one needs a measure of the pace at which each racer is moving, so, for a unit of time a unit of length needs to specified for each racer indicating the distance he travels. For unit change in time the position of each racer changes by his particular unit of distance and, since Achilles is the faster, at some point in time he will overtake the tortoise.

Is Zeno mocking Pythagoras' rule never to walk over a balance? Certainly Achilles would not be bound by this rule. With Zeno's partitioning of the race course the balance point is never reached. Is his purpose to indicate that a distance can be divided into arbitrarily small segments, i.e., that there is no lower bound for a unit of length? The steps that result are not uniform and there is a lack of a sense of the elapse of time. Aristotle discusses the role of the number line in his Physics and also points out that motion takes place in time.

One must also ask if Zeno's paradoxes are mnemonic devices, aids to the memory, in a time when not everything was written down. But that does not seem to be their purpose. For an explanation of motion their emphasis is wrong. It is more likely that the paradoxes were intended as topics for discussion to promote critical thinking.

The distances that Achilles has to cover to reach the tortoise's former positions are related to each other in a manner similar to that of octaves which are produced by halving the length of a string. Is this Zeno at play?

There is possibly another explanation for the need to approach a position as a limit. At about this time Antiphon, Zeno's contemporary and another supporter of Parmenides had proposed a method for the determination of the area of a circle which involved filling the circle with inscribed regular polygons with increasingly greater number of sides so that the area of the circle would be approached by exhaustion. The paradox may have been intended to provide support Antiphon's argument. It is possible that they met when Parmenides and Zeno went to Athens which according to information in Plato's Parmenides was about 450 BC. Plato's Parmenides is purported to be a recollection by Antiphon of a discussion between Socrates, Parmenides, Zeno, and Aristoteles about one and many, whole and part, and motion and rest.

I suppose that what I was trying to do is include some critical thinking in the article but apparently that is not the place for it. I should note that perhaps it would be better to include the other Aristotle paradoxes under some other heading if the wish is to restrict the article just to Zeno's. Would a link to the list of paradoxes be out of order? --Jbergquist 17:15, 24 June 2006 (UTC)
Those are not Aristotle's paradoxes they are Zeno's. Paul August ☎ 21:19, 24 June 2006 (UTC)
  • I think it need not be made any more complicated than the rationale given in the intro. Anything else might be interesting, but would have WP:OR issues --JimWae 19:20, 24 June 2006 (UTC)

[edit] Uncyclopedia link

User:Monkey 05 06 recently added:

For those with comedic tastes, Uncyclopedia, the Wikipedia parody site has an article about Zeno's Paradox.

Is this appropriate? —Keenan Pepper 03:35, 24 June 2006 (UTC)

Perhaps if his judgement about how funny it is was not coloured by his having written it --JimWae 03:38, 24 June 2006 (UTC)

Well, I don't really know if Stbalbach has any authority to make these types of decisions, but on the discussion page of the banned Uncyclopedia template, he clearly states:
"No ones stopping you from linking to a Uncyclopedia article as an external link, just not using a graphic banner" (Stbalbach 16:19, 1 February 2006 (UTC)).
This is just one example of several statements in the discussions stating that textual links to Uncyclopedia are okay.
And the fact that I wrote it hasn't created the desire to link to the article, it has just fueled it. Personally I feel that letting people know that someone wrote a parody article of serious subjects isn't a life or death matter. I don't have any power over this site though, so it's not my place to say. However, others who do have a right to make these decisions have said it is okay, so I am going to exercise the rights I have. I have every intention of placing more links, I just placed this one first because, as you say, I wrote it. However, that shouldn't have any effect on whether or not it is okay for the article to be linked to. Monkey 05 06 03:48, 24 June 2006 (UTC)
I think you misinterpret Stbalbach's comment. He didn't mean links to Uncyclopedia are always okay, just that they might sometimes be okay, as opposed to the big box which could never be okay. I don't suppose you'd be satisfied with putting the links on the talk pages instead? —Keenan Pepper 03:54, 24 June 2006 (UTC)
If the link is placed on the talk page the majority of the article readers will not see the link. Being the author of the article I'm sure that this makes me look very much like some type of attention whore, however, the part that upsets me is that no one seems able to agree as to whether or not this is acceptable at all, and so when someone takes action on the matter everyone jumps my case. I posted one link. So, at least in my opinion, that would qualify more along the lines of "sometimes" than "always". If you don't want me to post more links, fine, I won't. But Stbalbach said this was okay. Others seem to think it's not. It's one link at the bottom of the page. The majority of the reader's probably won't notice anyway...I don't see why it's such a huge ordeal anyway.
And one further thing JimWae, I didn't post the link because I would rate the article as being the funniest thing in the world. I'd rate it as moderately funny, perhaps a 5 or a 6 on a 10 point scale, but no higher than that. And if I'm really being overly biased with that rating then I apologize for being overly prideful. Monkey 05 06 04:08, 24 June 2006 (UTC)

There are lots of "zeno" links we could have. We need to be selective in including links, and I don't think the "unencycopedia" link is particularly informative. Paul August ☎ 04:21, 24 June 2006 (UTC)

[edit] Alternative statement of Achilles

In the Stanford Encyclopedia article on Zeno's Paradoxes there is an alternative statement of Achilles and the Tortoise attributed to Simplicius with editorial insertions. Aristotle's comments on Zeno claimed that he was wrong. Aristotle is not a primary source for Zeno's paradox and is somewhat biased against him and Simplicius's statement may be more accurate. A different set of rules should be applicable for secondary sources. Perhaps this could be reviewed and the article modified accordingly. --Jbergquist 21:30, 25 June 2006 (UTC)


[edit] The story's "Givens" create the paradox

It is possible for the tortoise to win the race; this would occur if during the time it takes Achilles to reach the tortoise's starting point, the tortoise has already finished the race. This is possible, but according to the story: It is given that the tortoise will always furthur advance; therefore, the story defines a paradoxicial situation: the (slower) turtle will always advance ahead of Achilles (who is faster); and will do so infinately (with no limit or stopping point). To restate what has been stated: "To finish a sequence that has no finish is a logical contradiction."

Question.... how is it possible for a slower body to infinitely move ahead of a faster body?

I think a good answer to this question/story would be: Mu

[edit] The Dichotomy Paradox

There are an infinite amount of "ways" to measure something finite; but that does not mean that what you are measuring is infinite. I could break the "distance B" into 1 million tiny fragments of equal length. yet when I add these fragments together, I will be left with the "distance B". I could instead break the "distance B" into eight equal subdivisions, yet when I add these fragments together, I would again be left with the "distance B". And I could define "distance B" much like I define the word "ocean" or "mammal."

Likewise, if you were able to measure tiny moments of time, you could divide 16 minutes and 40 seconds into 1000 seconds. If you like, you could divide 16 minutes and 40 seconds into 1000000 milliseconds.... You could "theoretically" divide even further, defining smaller and smaller subdivisions...but the human being would be unable to experience such "distinct" subdivisions, we are too "big" to recognize such tiny fragments as distinct and meaningful. But that doesn't mean we cannot experience a "second" or what we call a "minute" of time.

We may not be able to recognize a millisecond, but we can recognize a second; and intellectually, we understand what is implied by a millisecond...intellectually we understand that a millisecond is just one thousandth of a second.

[edit] Newton and inertial frames?

This is out of my ken but do you think it would be relevent to mention the concept of relative inertial frames / Newton's idea of a lack of an absolute state of rest?

In space, where there is no absolute state of rest, any of the following are equally "true" :

1. Achilles is stationary, and the tortoise is moving toward him at a speed equal to Achilles' ground speed minus that of the tortoise.

2. The tortoise is stationary, and Achilles is moving toward the tortoise at a speed equal to Achilles' ground speed minus that of the tortoise.

3. Achilles and the tortoise are moving towards each other at the same speed (each at half Achilles' ground speed minus the tortoise's ground speed) or at different speeds (any of several possible combinations of speeds totalling Achilles' ground speed minus the tortoise's ground speed)

For example, in the third example (where Achilles and the tortoise are moving towards each other) the idea that "the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead" is no longer applicable.

And how about relativistic time dilation? What bearing would this have, if any?

I leave it to someone with a better understanding of these matters to decide whether it is worth mentioning. (Probably the paradox of motion is still "valid" in pure space, requiring just a different presentation of the problem.)

[edit] Delisted GA

A single broad reference cannot possibly make this article "well-referenced" folks. You've got to pin this subject down, and good references should be able to do just that, but without it, you're bound to get just a bunch of people's private opinions about this thing, and it'll get confusing. Homestarmy 17:07, 14 September 2006 (UTC)

All of the external links are also references and, like many mathematics- and non-controversial science-related articles, this is not a collection of private opinions. —Centrx→talk • 00:30, 19 October 2006 (UTC)
Then why are they in the external links section, instead of the references section? Homestarmy 21:13, 27 February 2007 (UTC)

[edit] Uncertainty Principle

I wonder in how far this is connected to this paradox - and more, why it is not even mentioned... 212.34.171.12 10:16, 25 October 2006 (UTC)

It is now (mentioned :). Best regards, Steaphen 22:29, 6 December 2006 (UTC)

[edit] Objective Speed

I understand the basic thought behind the paradox, but it seems to me that the only reason it seems true is because you never attach an objective speed to either Achilles or the Tortoise. It seems like Achilles' distance is only ever measured in relation to the tortoise, or vice versa, and not against any objective constant speed.

And if we were to say something like, "The tortoise moves half the speed of Achilles," couldn't we also say, "Achilles moves twice the speed of the tortoise?" If the tortoise had a ten-meter head start, Achilles would run two meters for its one and would overtake it at the twenty-meter mark. It's as though if we move Achilles first and the tortoise second, the tortoise can never be caught, but if we move the tortoise first and then Achilles, it could easily be caught. Jboyler 17:55, 26 October 2006 (UTC)

[edit] Recent infinite divisibility additions

Adding people who agree with a certain interpretation is not sufficent to have wikipedia state that such an interpretation is factual or the only possible interpretation. POV need to be identified as such, not just present sources that agree with it --JimWae 19:49, 5 November 2006 (UTC)

Despite additions made by the originator of this new "stuff", the above request has not been met. The new additions are clearly POV & likely Original Research. On those grounds alone they merit deletion. The POV is also not well explained, just hinted at. More needs to be done by the originator of these additions to show any of it deserves to stay in an NPOV article --JimWae 03:06, 6 November 2006 (UTC)

[edit] questions for JimWae

JimWae,

Could you clarify your concerns?

Also, I have posted some material [[1]] (on my talk page which may assist).

As for NPOV (the heading to the section 'Are space and time infinitely divisible') ... teh supporting quotes give strong argument towards answering that question, so I'm unsure under what circumstances any provision of material is not a POV.

Also, I preface the beginning of most paragraphs with "if space-time is discrete ..." then various quotes from various physicists are used to clarify this position.

Steaphen Steaphen 07:58, 6 November 2006 (UTC)

[edit] removed paragraph in section "are space and time infinitely divisible" + included add. ref.

Previous material in the section "Are space and time infinitely divisible" included the following paragraph:

"whether or not space and time are measurable with infinite precision is ultimately irrelevant to the paradoxes and their resolution: what we as humans can know about the world is a different matter from what is or is not true or possible in the world."[citation needed]

I updated this with teh preface "Some have argued that "whether or not ..."

I've now removed the entire paragraph as it attracted the comment that the section lacked appropriate references.

Hope this is in accord with wiki procedures.
Steaphen 20:58, 6 November 2006 (UTC)


  • A detailed discussion thread on the subject of "Are space and time infinitely divisible" is included here (Steaphen's Talk page)

[edit] Re: Status of the paradoxes today

Reference is made to the fact that "no engineer has been concerned about them ..."

This appears to be unsupportable, particularly as there has been at least one engineer at Princeton University(PEAR) Robert G. Jahn, (past) Dean of the School of Engineering and Applied Science, who has investigated the finer paradoxical implications of quantum behaviour.

There are articles written on the inherent paradoxes of life (and consciousness) at [this site]. In the quantum realm the mind-matter interconnection is not resolved (to whit, the Schrödinger's cat dilemma) and is arguably not entirely unrelated to the resolution of Zeno's Paradoxes, as it raises the issue of how quantum behaviour transforms into macro-world Newtonian behaviour.

Regards,
Steaphen 22:27, 6 November 2006 (UTC)

[edit] questionable section in "Are space and time infinitely divisible" removed

The paragraph that read: "In a similar manner to Newton's laws of motion being approximations to finer quantum and relativistic principles, the infinite series that underlie most of the paradoxes are therefore geometric approximations of deeper super-luminal (nonlocal) "processes" or connections.{{POV-section}}" was removed to allay concerns of a POV being espoused.
218.214.81.20 06:37, 7 November 2006 (UTC)

[edit] Some ideas about the paradoxes

Jake95 18:53, 30 November 2006 (UTC)

[edit] The tortoise one

It must be remembered that after Achilles arrives at the spot the tortoise started, the next relevant point would be only one foot away, then the next \frac{1}{100}ft away, and so on, and the distances become so incredibly small quite quickly. So when is catching up? Is it being level with the opponent? If so, how can you measure that if the distance between them quickly becomes \frac{1}{10000}ft? It's practically impossible. Thus although I cannot disprove it, I think no-one can prove it. At least if they were not trying to do so mathematically.

Contrary to myself, he should never overtake because if you carry on dividing by 100 each time (which is what is happening) you will never get a negative number, which is required for Achilles to overtake.

[edit] The dichotomy one

But the distance to run is the total difference from where Homer is to the bus, is it not? Splitting it up is unnecessary.
Anyway, the first distance to run is \frac{1}{\infty} of whatever unit of distance, seeing as you cannot get larger than infinity, surely?

[edit] The arrow one

If movement is impossible, then eating is impossible. Therefore everyone would starve to death. Suggesting that life itself is an illusion.

Plus Aristotle says 'If' everything when it occupies an equal space is at rest. If it is, then it's at rest in midair. It should therefore fall to the ground.

--


[edit] Reply to above (and a reply in general):

To discuss or give opinion on the deeper issued behind Zeno's Paradoxes without reference to, or understanding of, quantum theory and experimental quantum research seems to me to be a very shallow approach to the problem.

For me, it is akin to discussing where the edge to our flat Earth might be, all the while ignoring the evidence that suggests the Earth is actually round; or believing that the Earth is the centre of the universe, despite the concepts advanced by the likes of Copernicus and Galileo who argued otherwise based on the observed facts.

We are now at a juncture similar to what we might have observed if we'd lived in the time when the Earth was believed to be flat, or the centre of the universe.

The evidence of quantum physics is compelling. It behooves those of us who seek to understand the deeper nature of reality (and the "solutions" to Zeno's Paradoxes) to acquaint ourselves with the findings of quantum mechanics.

Otherwise, it is just as meaningful to discuss how many angels might fit on a pinhead.

Steaphen 22:55, 6 December 2006 (UTC)

[edit] Quantum mechanics section garbage

All the mentions of quantum mechanics here read like an overeager high school student who just picked up a few new vocabulary words wrote them. What, exactly, do the uncertainty principle and Bohr's principle of complementarity have anything to do with Zeno's paradox? This is New Age claptrap and balderdash, not science. Someone who actually understand quantum mechanics (and some of the situations where QM DOES apply to Zeno's paradox) should write these sections.

The mentality displayed here, and in the article, appears to be one where QM is treated as some mystical, incomprehensible thing that somehow magically explains everything, which simply is not true.

In short, unless someone can give a more concrete explanation of how QM applies here, I move that these sections be removed, as they're not scientifically accurate and certainly don't befit an encyclopedia article.Egendomligt 09:27, 22 December 2006 (UTC)


-- I agree. I think it would be useful to have an informed discussion of quantum physics in the article, particularly in regards to the possibility of discrete space/time suggested by some theories of quantum gravity (such as Quantum Loop Gravity). However I dont have the required knowledge to write it, and an article filled with pseudo-science isnt a good compromise. - Gordon Ross 09:50, 22 December 2006 (UTC)

Re: "an article filled with pseudo-science"

Please qualify your use of the term "Pseudo-science".

In regards to the use of calculus in any solution to Zeno's Paradoxes, it is clear that such models do not work for physical movement at small spatial measures.

So why would you wish to use a (mathematical) tool that clearly does not work?

If calculus can be shown to resolve the Uncertainty Principle (ie. to give accurate description of momentum and position of quantum particles in Zeno's arrow), then well and good, and you will be rewarded with a number of Nobel Prizes for your efforts.

-- I don't quite understand why it's important to give position and momentum simultaneous values in Zeno's paradox. You keep saying that QM confounds the issue but it is not clear to me why it does.

In the meantime, I wonder who has engaged "psuedo-science" ... ideas or models that have little or no foundation in, or congruency with, physical reality.

[edit] A Disconnect between theory and fact?

As the author of those sections removed, I'm curious as to how quantum physics does not apply to the small (infinitesimal) measures cited as the foundations of calculus.

It seems to me that for those who argue quantum PHYSICS has nothing to do with solving a very profound PHYSICAL problem, there is a disconnect between some hypothetical idea of how to resolve Zeno's paradoxes, and with observed (quantum) facts.

-- I don't doubt that QM has interesting implications for Zeno's paradox. However, this is probably more a result of the wave nature of matter, the inability to "pin down" a point in space as being an object's distinct location, and so on, rather than any nonsense about time and space being quantized (for example).

The quantum facts speak for themselves, and are very much applicable to the solution of Zeno's Paradoxes.

If you want, I'll give far deeper explanation of why that is the case.

-- I'd like to see that, because the article isn't exactly clear on it and I feel it's better to leave these sections out until it's crystal clear what you're referring to. Feel free to revert it if you disagree, but you see where I'm coming from.

In the meantime, it would be courteous of you to repost those sections.

Thanking you in advance.

Best regards, Steaphen 01:06, 23 December 2006 (UTC)

Author, Awkward Truths: Beyond the Dogmas of Science, Religion and New Age Philosophies.

ps. If we wish to be disciplined with only posting concepts or information that is verifiable and impartial, then we will be expected to remove all mathematical descriptions that purport to resolve Zeno's Paradoxes -- particularly any reference to geometric series/Calculus, as such mathematical tricks have been unequivocally demonstrated (via quantum theory and experiment) to fail in describing movement of quantum particles (which comprise the physical stuff of arrows and hares).

If proof exists of how calculus and geometric series can resolve the uncertainty of momentum and position issues in quantum mechanics, please post, otherwise removal of all mathematical descriptions which do not concur with physical (quantum) facts should be actioned as a matter of priority, lest Wikipedia gain a reputation for silly and unqualified conjecture.

Once again, to suggest that the most successful physical theory in history has no relevance to a paradoxical physical phenomenon is quite odd, to say the least.

Such denials brings to mind the hysteria that we might have observed around the Salem witch trials, or the denial of the heliocentric model (espoused by Galileo et al) when offered in response to the physical facts.

Let's be clear: the geometric series "solutions" to Zeno's Paradoxes are reliant on assumptions of perfect space-time/physical continuity. There is no proof of such. Indeed, by way of quantum theory and experiment there is ample proof to the contrary.

What proof is there of perfect continuity of space-time in order to justify the applicability of the theory of limits/calculus/geometric series to resolve Zeno's Paradoxes?

Steaphen 01:06, 23 December 2006 (UTC)

-- What evidence is there to suggest that space-time is discontinuous?

//reply by Steaphen: If, as physicists David Bohm wrote in his "pop-science" book that movement is not continuous, then what happens in between those jumps? Once again, if there is no evidence that space-time is perfectly continuous, what justification do you have for using mathematical tools that are reliant on such continuity?

I hate to burst your bubble, but Bohm's interpretation of quantum mechanics is not generally accepted by physicists (nonlocal hidden variables and the "quantum potential" interpretation are incompatible with special relativity). You may yet nonetheless be correct, but that has no bearing on whether or not this line of discourse belongs in an encyclopedia article.

Irrespective of whether space-time is actually continuous or not, what evidence can you cite in order to justify a theory based on (at this stage) mere supposition?

If no evidence is available, what reasons do you have for assuming your suppositions are valid?

I note that the respondents have not answered this fundamental question. This is, for me, an illustration of something akin to "flat earth" thinking.


The question was Homogeneity and isotropy of space are two DEEPLY-held assumptions about the structure of the vacuum, and there's no reason, as far as I can see, to doubt them

//reply by Steaphen: similar "deeply held assumptions" were held in Galileo's day, about the Earth being the centre of the universe.

//reply by Steaphen: once again, what evidence do you have to support the use of a theory reliant on perfect space-time continuity?

This is NOT how burden of proof works. Burden of proof begins with a statement that is accepted, and the burden falls on the opposing party to counteract it. Another consideration is parsimony: there's no obvious reason to believe in some mystical mechanism by which space and time are miraculously quantized, so the simpler view - that they just aren't - is the one that ought to be favored. Essentially, you're calling me a flat-earther for asserting that fundamental classical assumptions which lead to silly things like conservation of momentum and energy ought to hold (when, again, there's no obvious reasonwhy they shouldn't).

- things like Noether's theorem rely on their usage. You cite (for example) things like the quantized energies permitted for electrons orbiting nuclei as evidence, but this is simply the nature of electron bound states, NOT a restriction on the nature of space-time itself. Egendomligt 08:52, 23 December 2006 (UTC)

Please use the preview function. You are flooding my watchlist. Dekimasu 03:34, 23 December 2006 (UTC)

My apologies. I was not mindful of the number of corrections I had posted. I'll be more diligent in the future.

Best regards, Steaphen


-- The problem isnt with including references to quantum physics, its that the sections I removed show a basic lack of knowledge about quantum physics (as evidenced by the fact that most of the cited references are pop-science books).

//reply by Steaphen: Interesting. "as evidenced by..." ??? So might I assume that "pop-science" books are entirely incorrect, in error? Okay now I'm getting a measure of the calibre of intellect of the people replying here. Thank you for these insights.

Space and time are not discrete in standard quantum physics, and the possibility of a discrete space is (afaik) one of the more interesting aspects of speculative theories like LQG.

//reply by Steaphen: such ideas are "speculative," presumabably because there is no evidence or experimental results to support the theories. Which again brings me back to the basic problem of using calculus to solve Zeno's Paradoxes. What evidence do you ahve to support the use of a theory reliant on assumptions that have not be shown to be valid?

Calculus has not 'failed' in quantum physics. Heisenbergs uncertainty principle has no obvious relevance to Zeno's paradox.

//reply by Steaphen: Physicist Fred Alan Wolf wrote: "Werner Heisenberg was awarded the Nobel prize in physics for his realization that Zeno was correct after all." Whilst Fred Alan Wolf (past Professor of Physics at San Diego State University, and recipient of the National Book Award for his book "Taking the Quantum Leap") might write "pop-science", I see no reason for you to imply that he's an idiot. The ideas can still be valid.

What's the rest of that quote, anyway? The argument seems to be "uncertainty principle => space/time discontinuous" and I've actually seen this argument numerous times, but there isn't any reason to suspect that's actually the case.

And so on. The whole section is just a rambling mess that has only a tenous connection to the article.

//reply. "tenuous connection" Again, "Werner Heisenberg was awarded the Nobel prize in physics for his realization that Zeno was correct after all."

What is tenuous about that statement linking quantum theory with Zeno?

Try to find a paper about QM and Zeno thats been published in a respectable journal (physics or philosophy) and cite that, instead of writing original research based around a large number of loosely connected quotes from popscience books. -- Gordon Ross 10:38 23 December

//reply. Why? My question was presented, but remains unanswered. What evidence is there to support the use of theories that rely on unproven assumptions?

Once again, if there no evidence forthcoming, I see no reason to not delete all entries that mention calculus or geometric series.

But hey, it's Christmas, so I'll leave this forum with good grace ... for the time being. But I think a New Year's resolution should be to clean up this forum with removal of theories based on unfounded conjecture and assumptions.

Maybe the Santa fairy will bring you the evidence to support your assumptions. :)

Completely agreed, as someone who's also trained in quantum mechanics. Egendomligt 19:06, 23 December 2006 (UTC)

//reply. Well then, you must be absolutely right ... and can advance a theory to show how your assumptions are not merely speculative, but evidentially based. I await your reply, and I'll be amongst the first to congratulate you on receiving the Nobel Prize, which will be the inevitable reward for your efforts. Most certaintly! :)

I'm sorry, but again, burden of proof lies on the person making the claim - in your case, you! I work with vacuum structure theorists on a fairly regular basis, and what most of these assertions regarding space being discontinuous have in common is that there's no a priori reason for them to be true and they're untestable... Look, I get the feeling a lot of the dischord here is coming from the fact that we're using the same words to mean different things. There's a terminology divide. If you want to write a section that uses the terminology properly on how quantization of space leads to a new view on Zeno's paradox, one that doesn't read like a grab-bag of QM-related terms - or point us in the direction of some concrete sources - by all means go ahead. 216.183.74.59 00:35, 24 December 2006 (UTC)

To summarize the problems:

1) There is nothing wrong with popscience books, however they should not be cited in an encyclopaedia as primary evidence for a scientific theory. When you read something in a popscience book, it can be hard to tell the extent to which what youre reading is the author's personal beliefs, rather than widely accepted/demonstrable fact. Ok, so you have a quote saying that David Bohm thinks X. What of it? Physicists disagree about lots of things and its not particularly difficult to find very intelligent, educated and respectable figures espouging fringe positions. David Bohm is hardly representative of mainstream interpretations of quantum physics and while he may very well be right, his views should not be cited in an encylopaedia in a context which implies that they are widely accepted.

2) (Following on from above) The section which you wrote uses assorted pop-science quotes to try and piece together a long narrative in an uncritical manner. This is a blatent example of original research.

3) A fair amount of what youve said is just wrong. I will repeat again - space is not discrete in standard quantum mechanics. Neither is time. This situation may change in the future (for example if LQG becomes widely accepted), but at the moment space and time are generally considered to be continuous. Yes, you may be able to find individual scientists who believe otherwise, but the purpose of wikipedia is not to report on every single viewpoint held by every single figure in the field - it would be unmanagable if this were the case. Theres nothing wrong with writing a few sentences noting that 'several scientists believe <non-mainstream position X>', but a huge section devoted to fringe science like the one you wrote is completely irrelevant to the article.

4) You still havent explained why calculus 'fails', or why Heisenberg's Uncertainty principle is relevant to Zeno's paradoxes (other than providing a contextless quote). Do you honestly think that theres likely to be some new theory of physics which doesnt use calculus or differential equations?

5) In the context of wikipedia, the burden of proof always lies with the person who is challenging mainstream science. This is an encylopaedia rather than a research journal - its function is to describe the current state of the world and academic research, not to propose and defend daring new theories.

6) I'm getting the impression that you havent studied quantum physics to any serious level. While credentials arent everything, I dont think that youre qualified to write a long description of quantum mechanics for wikipedia which flies in the face of mainstream science. This isnt intended as a personal attack - I would personally be reluctant to write something controversial in this article because I only have an undergraduate level knowledge of quantum physics. I would be perfectly happy for someone who knows more about the subject to write a section on QM and Zeno -- Gordon Ross 24 December, 05:24.

    • Reply to the above by Steaphen:

"This is a blatant example of original research .." Possibly, but my intent here is to question the accepted dogma concerning the use of calculus to resolve zeno's paradoxes.

You make mention this is an encyclopedia. Well, shouldn't that require the validation of the use of geometric series/calculus. See below:

Regarding the failure of calculus: The trajectory of an arrow can be described mathematically (e.g. a parabola) which allows one to calculate at any point along its flight the height of the arrow (in ideal conditions), its rate of acceleration etc. These are standard Newtonian principles that work well.

However, if one consider the movement of the arrow in a very small spatial interval (e.g. in the region of the Plank interval/time) then can you likewise calculate the exact position of particles and their momentum (i.e. can you therefore calculate the exact position and momentum of the arrow -- which is merely a collective of particles)?

The point is that the (commonly used) tools that are assumed to resolve Zeno's Paradoxes don't work at quantum levels. If you can show otherwise, please do so.

THe burden of proof is not mine, given that I did not introduce the idea of using a geometric series to describe motion through quantum intervals.

No one has shown how standard classical physics (via geometric series/calculus(reliant on continuity of movement)) can apply in the small spatial regions through which the arrow must eventually travel.

I introduced material to counter the groundless assumptions concerning the use of geometric series to "resolve" Zeno's Paradoxes. However, if you want, then by all means remove my posts, but show some discipline and do likewise with the geometric series material.

This is elementary stuff. Why the difficulty in understanding this?

Since this is an encyclopedia, then by all means state the problem (as posed by Zeno) but including geometric series to "resolve" it is invalid, on that basis. The fact is that Zeno (or Aristotle) made such observations. The work surrounding the attempted "solutions" is also fact. But their validity is not fact. They are only assumed to work, and they're assumed to work via highly questionable reasoning.

"I'm getting the impression that you havent studied quantum physics to any serious level" .. Quite possibly, but I fail to understand the relevance of this comment. I studied level 1/advanced physics at high-school and at undergraduate level. The concepts concerning the movement of physical objects shouldn't need a doctorate to understand. If they do, I'd say there was be something fundamentally wrong with your concepts.

Once again, no one has shown how standard classical physics (via geometric series/calculus(reliant on perfect continuity of physical movement)) can apply in the small spatial regions through which the arrow (and the hare, tortoise and runner) MUST eventually travel ... or do they somehow "jump" the quantum realm?

As for the use of terminology. This is largely irrelevant to the discussion here. We are talking about arrows, runners and hares etc. Physical objects. In quantum physics, mathematically, multi-dimensional "space" (e.g. Hilbert Space), Heisenberg's matrix mechanics etc assume or work with dimensions and variables that may not be tangibly physical. According to some interpretations (e.g. Everett's Many Worlds interpretation), those other dimensions are real, but just not real to us, in this reality. The point is that in this dimension (or physical reality) calculus fails to continue to work at the quantum realm. That's my point. We have to introduce quantum mechanics to "follow" or describe the particles (and thus the arrows). But if those variables/dimensions are not tangibly/physically observable, what does that tell you?

Failing any proof to the contrary I suggest the references to geometric series be either removed, or put in the correct context: that they're speculative, groundless theories that are, to date, not supported by experimental evidence.

[edit] 4 significant problems with Steaphen's edits

1. Neglect of NPOV in the use of terms such as "has been shown",

comments removed due to incorrect format, as pointed out by JimWae. However, Jim, I suggest you also discipline those who originally injected commentary into my replies. See above for replies to the latest responses. Plus, see below re "do your homework" [User:Steaphen|Steaphen]] 04:11, 26 December 2006 (UTC)


"in fact", "Another solution",

    • I have never stated anything was a solution - only you have - and have tried to make the article say so too - and that is the fundamental problem with what you have added --JimWae 03:20, 24 December 2006 (UTC)



      • Nor did I. "Another solution ..." was already the introduction to the section "Are space-time infinitely divisible?" ... that was how I found it. It would be courteous of you to do your homework.

"Ipso facto", "has pointed out",

"As Morris goes on to explain", "As physicist Fred Alan Wolf explained", "More directly, Cahill noted that:" "perfectly smooth movement is... an illusion, as Zeno originally posited" "As physicist Fred Alan Wolf wrote...", "At Cambridge University, mathematician Noah Linden and physicist Sandu Popescu have found that..", "As physicist Nick Herbert explained", "As Oxford University physicist David Deutsch explained", and far too many more --- There is entirely too much endorsement of one POV.

  • the formation "as explained by.." indicates an endorsement of the opinion that follows - violating WP:NPOV--JimWae 04:41, 24 December 2006 (UTC)
  • you have mucked up the format on this page - please learn how to respond & still keep other people's comments intact. Ever read about the 4 tildas? --JimWae 04:43, 24 December 2006 (UTC)

2. Zeno's paradoxes are not about continuity or discontinuity of matter, but about space & time

3. "affirmed theoretically" is meaningless

4. much of the exposition is meandering --JimWae 06:23, 23 December 2006 (UTC)

However, the "jumping" of electron orbitals is relevant to the issue of motion - though how it is to be presented needs more work --JimWae 06:48, 23 December 2006 (UTC)

Agree, and I also heartily agree with the latest deletion [2]. That whole section is full of quotes taken out of context, most of them completely unrelated with this topic.

This is one of the most biased articles I've seen in wikipedia, and it has nothing to do with politics! Itub 15:16, 23 December 2006 (UTC)

Also relevant to the article, though by no means the final word, are Planck units - which are often erroneously defined as the smallest meaningful units rather than the smallest measureable differences. --JimWae 05:20, 24 December 2006 (UTC)

  • Nobody gets to assign homework to anyone here. If I mistakenly commented on edits that were not yours, just say so
  • Stop deleting comments from this page - it makes it especially hard to follow a thread
  • learn & use the 4~s --JimWae 05:32, 26 December 2006 (UTC)


[edit] Zeno’s Paradoxes, Wikipedia and Apple Carts

As many of you would be aware, there are many many blogs and forums that discuss what many here (at Wikipedia) would see as unsupportable, biased opinion.

This encyclopedia (Wikipedia) is supposedly about presenting the facts and – at least as far as this section on Zeno’s Paradoxes is concerned – credible theories that support or explain those facts.

With regards to Zeno's Paradoxes, they involve the issue of movement through increasingly small physical intervals.

One example used by Zeno was the flight of an arrow.

Part A:

The arrow moves through physical space in a trajectory in accord with Newton's laws (e.g. in a parabolic curved trajectory). At each point along that trajectory, mathematically we can say at point 'x' along its path, the arrow will be at 'y' height. Moreover, at any point along its path we can mathematically determine its physical characteristics of position, momentum and rate of acceleration (given known initial conditions).

The variables in describing the flight of an arrow all have physical attributes (e.g. momentum, position, rate of acceleration etc.).

Let’s imagine (a thought-experiment) in which we fashion ourselves a particularly fine arrow (one so fine that its tip has been honed to just one iron atom). Since, according to classical physics we can accurately determine the arrows trajectory, we can likewise accurately determine the position of the atom.

Lets imagine we fire the arrow in a complete vacuum, so as to not to confuse the issue with friction losses, cross winds and other physical influences.

According to classical physics, at any time during its flight (assuming known initial velocity, weight of the arrow) we can apply classical equations to physically predict not only the arrow’s (and the atom’s) position, but also its momentum.

Note: we were particularly diligent and analysed the arrow to determine not only its weight but also the number of atoms it contained, thus enabling momentum for the lead atom to be determined.

When geometric series are used to “resolve” Zeno’s Paradoxes, the assumption is made of perfect continuity – that is, there are infinite points along its path.

Accordingly, at any point in time, we can precisely determine the position of the arrow’s lead atom, and its momentum.

Unless there is something wrong with my reasoning, this means we’ve busted the Uncertainty Principle of quantum theory. Nobel Prize please.

Part B:

Assuming the reasoning in Part A is sound, we obviously cannot precisely rely upon classical physics to map the progress of the lead atom in the arrow (and in a more general sense, the arrow itself). We may invoke the use of quantum theory in order to give very high (almost certain) probability of physical attributes of the lead atom.

However, quantum theory is reliant on variables that do not have associated physical attributes ... in other words, the equations mapping the (highly) probable nature of the arrows path no longer have direct and tangible equivalence in (this) physical reality.

I am sorry but this is flat-out, demonstrably wrong. Nothing in QM relies on non-physical attributes, or attributes which cannot be in some way measured. In fact, QM states that no variable takes on a definite value until it is measured. To proclaim that a "non-physical" variable is necessary is absurd.Egendomligt 02:46, 28 December 2006 (UTC)
No need to apologize, other than to explain how strict determinism (assumed when using geometric series to resolve Zeno's Paradoxes) is still true when the arrow traverses through those quantum intervals. As for your "QM states that no variable takes on definite value until it is measured ..." it is exactly that fact which distinguishes quantum theory from deterministic, classical physics. What is the reason for the variables not innately having a physical value before it is measured (which is assumed in classical physics)?
Look, no matter how much you avoid the central issue, the fact is you cannot justify the use of geometric series to resolve Zeno's Paradoxes.
In classical physics, until we "measure" the height of an arrow as it flies through the air, it is still assumed that it is physically "up there". This is not the case for QM. As you may be aware, this has led many scientists to question the deeper nature of reality. Einstein refused to accept QM quipping that ... words to the effect of, "I don't accept that when I'm not looking at the moon, it isn't up there." Whatever that reality, at present QM says it cannot be precisely determined.
And by the way, JIM! ... what's with these interjections into MY posts ????????????????????


There is no direct and complete physical correspondence between the variables and physical reality. With quantum theory, strict determinism is no longer achievable or valid.

Now, with Zeno’s Paradoxes and the use of geometric series, the points in the series have direct physical correspondence (each point in the series is taken to be a point of position of the arrow, hare, runner etc.)

Accordingly, the use of geometric series to “resolve” Zeno’s Paradoxes is incongruent with the actual, finer physical motion of the atoms that compose the arrow (and the hare, tortoise, etc).

As I originally posted, with the discovery of quantum physics, classical physics (and calculus) has been shown to be insufficient to accurately and reliably map or predict movement of physical matter in the quantum realm.

Cheers, Steaphen 05:39, 27 December 2006 (UTC)

ps. if you look to your equations (wave functions) I think you'll find that any continuity of space is reliant on non-physical variables. In other words, strict determinism fails. Our particular space is reliant on "other-space" (non-physical) variables. So if our terminology is about "our space" -- one that is physically observable, tangible and real, then no, I do not believe it is continuous. And the quantum evidence supports this. I don't believe Feyman's Sum Over Histories or Everett's Many Worlds interpretations etc assume continuity of OUR space, merely continuity of an infinite-dimensional "space" (a few dimensions of which are ours). All of which invalidates the standard geometric series/theory of limits treatments of Zeno's Paradoxes (for reasons detailed in Part B above).

Some physicists are working to advance more congruent frameworks of understanding, here at http://www.qpt.org.uk/

Steaphen 06:30, 27 December 2006 (UTC)

I am looking at every wavefunction I have ever seen and NONE of them mandate the existence of non-physical variables - indeed, the Bell inequalities demonstrate that such variable theories are incompatible with QM. Additionally, it is apparent to me that you do not understand what constitutes a "dimension" in physics.Egendomligt 02:46, 28 December 2006 (UTC)
    • Reply by Steaphen
It has been a while since I immersed myself in the mathematics, and my terminology may not be as correct as should be. However, again, the concepts that I am focused are as follows: in quantum theory strict determinism fails. There is no strict correlation between all variables and (tangibly) physical attributes.
However, in using geometric series to resolve Zeno's Paradoxes, strict correspondence is assumed. This is contrary to quantum theory. Since any arrow, hare or runner will progress through quantum intervals/measures, then one must apply quantum theory (or some successor to it). The use of geometric series is invalid.
Please show any evidence to prove or illustrate otherwise.
Thank you.

[edit] Zeno's paradoxes are mathematical, not physical

Zeno's paradoxes are nothing more than mathematical exercises, and as such they are solvable within mathematics. Saying that calculus can't be used because space is discontinuous is ludicrous, because the problem is abstract, not physical (it would be like saying that an area of a circle can't be exactly pi*r^2 because space is discontinuous). Whether physical space is continuous or not is certainly a valid scientific question which should be solved based on actual observations, but such information might be more appropriate in articles about space or vacuum. However, mathematical exercises by themselves cannot tell us anything about the physical world. If it happens that "Zeno was right" in his conclusion that space is discontinuous, that doesn't mean that his argument was right, any more than Democritus was right in the arguments for his atomic theory (atomic theory only became a solid scientific theory, based on many different kinds of observations, during the 19th and early 20th centuries.) Itub 15:58, 27 December 2006 (UTC)

    • reply by Steaphen.

re your "Zeno's paradoxes are nothing more than mathematical exercises" ... ???? Your response telegraphs an extraordinary disconnect of theory from observed reality.

Zeno questioned (essentially) how physical movement was possible. Using geometric series (calculus, theory of limits) is merely a tool to explain that movement. If it cannot relate to, or match physical reality, it is an inappropriate tool.

I suggest it be thrown away ... or amended/augmented to such an extent that it does work -- Voila ... quantum theory, not geometric series.

Yes, we get that you're saying this. That doesn't make it valid or true, though.Egendomligt 02:46, 28 December 2006 (UTC)

Using geometric series to "resolve" Zeno's Paradoxes is invalid.

Please be so courteous as to ammend the main article to reflect this.

Cheers, Steaphen 01:09, 28 December 2006 (UTC)


[edit] Neutrality and validity questioned.

I've inserted {{POV-section}} into the main article, and into the "Status of the Paradoxes Today" sections due to the above questions not being satisfactorily resolved.

The responses to my questions have indicated a bias towards dogmatic beliefs that are not validated by experimental evidence.

Best regards, Steaphen 01:52, 28 December 2006 (UTC)Steaphen

[edit] Put up or shut up

Steaphen, please provide journal articles which give serious evidence based on mainstream QM to doubt that space and time are continuous. You keep asserting that we need to look at the quantum evidence and that QM explains such-and-such but you never actually proclaim why this is so. Either put up or shut up. Otherwise, if nothing else, everything you've posted here is original research and thus can likewise be ignored.Egendomligt 02:46, 28 December 2006 (UTC)

Reply by Steaphen
The burden of proof (on the use of geometric series) is not mine to provide.
My material on the discontinuity of space-time is no longer an issue (since it has been removed).
Please explain why the use of geometric series is valid, particularly as I've explained in Part B above why it is resolutely NOT valid. This is not original research any more than questioning why someone who believes it is a "fact" that 450 angels can fit on a pinhead.
This is elementary stuff. What is the problem?
Once again, the burden of proof is yours to provide, otherwise remove the references to the geometric series material in the main article.
I'll again insert the POV challenge. If you remove it again I'll consider that to be case of vandalism.
Thank you.
You can "consider" it to be anything you like, but that doesn't make it the case. And I have stated before that this is not how burden of proof works. All my claim is is that there is no current reason to believe that a fundamental space quantum exists (one might cite the Planck length but this is not necessarily the same thing), nor a fundamental time quantum (again, the Planck time is not the same thing - unless you'd like to show me how it is!). Your claim is that there is reason to believe in such a thing. The burden is on YOU to provide some evidence for it.
You are multiplying entities beyond necessity - proposing that space-time is discrete when, as far as it has been experimentally tested (to my knowledge), it behaves like a continuum. Your assertion is both not parsimonious AND is not experimentally verified. Additionally, the position you are forcing on ME is indefensible. If I could provide evidence that space/time are quantized, you could simply fall back, for example, on the notion that our measuring instruments are not precise enough, and insist that there might still be a quantum beyond this resolving limit.
This is following the same pattern as a debate about the existence of God. Theist states God exists: atheist states God doesn't exist: theist pushes burden of proof on atheist: atheist cites burden of proof definition and Occam's razor: theist leaves conversation, knowing his demand on the atheist is not reasonable. It is time to accept that your position is not going anywhere.
I am reading Part B over and over again and it seems like the only argument you have made is that, based on the uncertainty principle, one cannot assign a simultaneous position and momentum (hence velocity) to the arrow. This is mitigated somewhat by the fact that, since the arrow is large, it behaves classically to a first approximation, but otherwise you are mostly correct. Nonetheless, this does not appear to actually pertain to the paradox: it's a nitpick, nothing more, and therefore deserves only a side mention. Again, if you can provide a source for this reasoning (so it is not original research), a small mention would be appropriate, it seems, but not the enormous, rambling behemoths you've left in the article before. kthxbye Egendomligt
05:28, 28 December 2006 (UTC)
Excuse me, but what has the discontinuity of space got to do with this dialogue? That concept was removed. It is not the issue at hand.
re your "there is no current reason to believe that a fundamental space quantum exists" Fine. Who cares? That's not the issue at hand. Please reread my Part A and B.
Let me spell it out. Geometric series rely on deterministic principles (1:1 correspondence between series points and points of physical position) which are not applicable or valid on quantum scales. If they were applicable we could dispense with quantum theory. We could determine exact physical attributes of matter, in all circumstances.
Once again, the use of geometric series to resolve Zeno's Paradoxes is not justified. If you have any reason to justify the use of geometric series (or any other deterministic tool) to describe movement on quantum scales, please enlighten me.
Steaphen 06:42, 28 December 2006 (UTC)

[edit] Mediation requested.

I'm thinking that given the demonstrated intransigence (over the lack of any validation for using geometric series to resolve Zeno' Paradoxes), that mediation (informal, then formal, if unresolved) be initiated.

I've not initiated such before. Suggestions welcomed. I'm short on time at present, but hope to investigate this matter later.

Steaphen 04:33, 28 December 2006 (UTC)
Yes, at this point it seems like mediation might be reasonable. Egendomligt 05:28, 28 December 2006 (UTC)
Request for mediation actioned.
Please refer here (not sure if html link is allowable).
Steaphen 07:41, 28 December 2006 (UTC)

I'm willing to mediate the situation, if the parties still feel that it is necessary. --JaimeLesMaths (talk!edits) 10:47, 27 January 2007 (UTC)

Is this case still active or can I close it? --Ideogram 21:27, 9 February 2007 (UTC)
Closing. If it needs to be reopened, leave a note on my talk page. --Ideogram 18:09, 12 February 2007 (UTC)

[edit] Summary of position

I've reread my replies above. Typically I write then reread once or twice, then post ... enjoying the spontaneity of the process.. : Anyway, I apologize for my loose (some might say, quite poor, or downright incorrect) use of terminology.

In summarizing my position, it is as follows (I'll try to be more diligent in my use of terminology):

The error with geometric series when used to resolve Zeno's Paradoxes is that geometric series are reliant on strict determinism (as stated previously, it is assumed that there is a strict 1:1 correspondence between the points in the geometric series and the physical points of movement (e.g. of the arrow, hare, runner etc).
We know from QM that such strict correspondence fails. In the replies above I said "non-physical" ... by that I meant that the variables have no strict physical correspondence to position, momentum etc. In a sense, they are no longer "physical" in the manner of an arrow flying through the air. The variables describing the flight of an arrow are still assumed (and expected) to represent actual, tangibly real physical attributes(irrespective of whether we get on a ladder and actually measure (catch) the arrow). This is not the situation with QM.
Given that an arrow, hare or runner must traverse quantum-scale increments, it is, I think, quite reasonable to ask how such a 1:1 deterministic correspondence can still apply when this has been shown (e.g. via the Uncertainty Principle) to not be the case (on such scales).
To suggest or argue that Quantum Theory is unrelated to the resolution of Zeno's Paradoxes (which is about the increasingly minute/small movements of runners, hares etc) requires an explanation of how the runner, hare or arrow moves (or jumps?) through those quantum-scale increments.
I hope my use of terminology now satisfies all. :)
If that be the case, I think my original inclusion of material (see below) should again be included (I concede this might require a change of wording, but the overall meaning and concepts remain valid).
As previously posted in the opening/introductory section(but was removed against my objections) -- with slight revisions:
"However, with the discovery of quantum mechanics early in the 20th century, classical physics (and calculus) has been shown to be insufficient to accurately and reliably map or predict movement of physical objects in the quantum realm.
At very small/quantum scales, the movement of physical things fails to conform to classical laws of strict determinism.
With quantum mechanics, new paradoxes have been discovered, such as Heisenberg’s Uncertainty Principle which prompted the thought-experiment of the Schrödinger's cat paradox. (See also Thomson's lamp paradox and Bohr's Principle of Complementarity)."
Is that better/more appropriate?
Kind regards,

Steaphen 13:37, 28 December 2006 (UTC)


  • You have not yet made a strong case for what to put in main article & yet you want to put 3 paragraphs on this one topic in the intro.
  • Thompson's lamp has nothing to do with quantum physics & everything to do with Zeno's paradoxes.
  • Calling Heisenberg's Uncertainty Principle a "paradox" is a stretch
  • the leap of electron orbitals is not the same as a leap of electrons, electron orbitals being the probability
  • the movement of macro objects has not "been shown" to mimic orbital leaps --JimWae 14:46, 28 December 2006 (UTC)
  • re the NPOV template added to intro - is there something in intro you object to, or is your objection completely about an omission. The intro must reflect what is in the main article - make your case for what to include in the main body first.--JimWae 14:51, 28 December 2006 (UTC)
  • is there a place where the article still says geometric series IS 'the solution. I made edits so that it is presented as a proposed solution. It is historically tied to Zeno's paradoxes & cannot be omitted just because you think it is not a complete solution. --JimWae 14:55, 28 December 2006 (UTC)


Perhaps I need to apologize. It seems I've not understood that Wikipedia is more about echoing accepted historical dogma, as opposed to presenting the facts, and solid theories to support those facts. I gave a link to a website where those theories are now being honed, compared, debated etc. If we are to present the facts without POV, then we should give equal mention to those as well. That has been my point all along.
Can I ask why you have mentioned "leaps of electrons"? What has this to do with the strict determinism of geometric series which are unable to be correlated with small movements of physical things (in quantum intervals)? Please reread Parts A and B (that reminds me, where's my Nobel Prize)?
re your "the movement of macro objects has not "been shown" to mimic orbital leaps" so? Please stay with the issue at hand about the invalidity of geometric series to resolve Zeno's Paradoxes.
Finally, re your "I made edits so that it is presented as a proposed solution." Well, isn't that a POV? I made contributions which can also be deemed a "proposed solution." Yet they were removed. I find your whole attitude very biased towards old dogmatic beliefs that (in the light of quantum theory) are no longer tenable.
if this article is to be without bias, then you would need to give far less mention of geometric series, mentioning them only in passing as reflecting old Newtonian conceptual views of our physical reality, which have been shown to be first approximations but are very superficial. Not withstanding the fact that they simply do not "fit" the observed facts (since we now know that strict determinism is no longer valid for very small/quantum increments of physical things). In very small increments of movement, geometric series have no foundation of use. Or is a "proposed solution" based on the number of angels on a pinhead as valid? (given that some of them might slip off the pin and end up floating around willy nilly, interferring with the arrow, deflecting it from its path, and causing all sorts of bother and mayhem). Come to think of it, is that why there's no strict determinism ...? Should I include that as a "proposed solution"?
re your "Calling Heisenberg's Uncertainty Principle a "paradox" is a stretch" ... you don't find it rather odd (paradoxical) that if we accurately measure an object's position, we have very little to no clue about its momentum (velocity)? Conversely, if we accurately measure an object's momentum, we then have no clue about its position. You don't find that paradoxical? As an analogy, that would be like seeing a Mack truck off aways, thinking "hmm, big truck, hurties if that hits me" then "Bam, wow, that hurt, didn't see that coming" because the velocity (momentum) of the truck was not able to be ascertained, and vice versa. Please, this is only an analogy, but you don't find QM paradoxical??? I agree with Bohr, that if you're not shocked by quantum theory, you've not understood it.
But let us not digress from the issues at hand: The inapplicability of geometric series as a credible foundation for resolving Zeno's Paradoxes. They should only be mentioned in a historical context, as being an outmoded conceptual understanding that is no longer congruent with our deeper understanding of physical reality.
Steaphen 20:01, 28 December 2006 (UTC)
btw, with regards to those who argue that Zeno's Paradoxes have nothing to do with Quantum Physics. It's precisely the opposite ... they have everything to do with QM, that's why they're paradoxes (because that's where paradoxes live in spades). ZP is a wonderful crucible to highlight how our Newtonian mechanical view of reality is wrong ... it's based on illusion. We only think OUR physical reality is continually here, but (as I quoted some physicists as saying, but the quotes were removed) it is flickering on and off (like a motion picture film) ... that's why things can't be tracked in a deterministic manner ... because in between blinks "they" cycle through the other parallel (non-physical/intangible to us) possibilies (hence the single-photon double-slit results), and the necessity for nonlocal connections, and ... and ... and ...Steaphen 20:18, 28 December 2006 (UTC)
Where are those mediators when you need them? :)
I have to agree with you that you don't understand wikipedia; specificaly the no original research policy. Itub 20:07, 28 December 2006 (UTC)
Moi, research? Golly I've barely done a lick of work my whole life. I merely reported what others have found (in quantum physics). Don't shoot me, I'm only the messenger. :) Steaphen 20:22, 28 December 2006 (UTC)
You collect quotations from quantum physicists and use them to formulate your own non-mainstream theory about Zeno's paradoxes. That counts as research too. Itub 20:27, 28 December 2006 (UTC)
That's your opinion. Please see below (Steaphen 21:02, 28 December 2006 (UTC))
Actually, no, it's not just his "opinion". It's specifically cited on the WP:NOR page as an example of original research. Egendomligt 22:02, 1 January 2007 (UTC)
It is possible to annoy me, and I feel that you're starting to succeed. I've enjoyed the exchanges, but now I'm losing interest and patience. The link you provided includes this quote: "any facts, opinions, interpretations, definitions, and arguments published by Wikipedia must already have been published by a reliable publication in relation to the topic of the article. " Once again, reread my posts .. the ones that you lot deleted due to your bias ... I simply quoted physicists on the issue of Zeno's Paradoxes, and their opinions as to the implications of physical movement at quantum scales (as espoused by physicists!). As I recall, you either mentioned or implied that "pop-science" books and the opinions and interpretations by physicists are not reliable or credible(publications). I'm wondering if some of those physicists (and their publishers) now have good grounds to sue you. What exactly is your problem? Steaphen 04:48, 2 January 2007 (UTC) Actually, yes, when I think about it I'm now annoyed. I'll be bumping this article to a request for formal mediation. Thank you.
I've annoyed you? Oh, Lordy, no! You had me when you conceded that your argument no longer revolved around quantization of space and time. That's fine. But this doesn't change the remaining fact a) that the rest of your argument is still a horrible rambling mess, and b) that the rest of the edits you've inserted are nothing more than a cobbling-together of out-of-context quotes with little or no material linking them. The ONLY relevant quotes I can find in your most recent large edit are the first Bohm quote for which no context is given, the Friedman quote which is irrelevant as, again, mainstream QM doesn't mandate the quantization of space and time, your Wolf quote about Heisenberg for which, again, no context is given, and... actually, that's it. Probably 70% of it is irrelevant to the paradox (and note that you've repeatedly refused requests to rewrite it in a more concise, explicit form).
I BESEECH you, please read the NOR page, specifically these two tabs: "It introduces an argument, without citing a reputable source for that argument, that purports to refute or support another idea, theory, argument, or position" and "It introduces an analysis or synthesis of established facts, ideas, opinions, or arguments in a way that builds a particular case favored by the editor, without attributing that analysis or synthesis to a reputable source;" as well as some of the things Jimbo has written on why Wikipedia relies solely on established, accepted secondary sources. Wikipedia is not a place for original ideas. This talk page is also not a place for serious scientific debate (except where relevant to what belongs in the article) or for aggravating each other (to which, I'll freely admit, I've contributed). Egendomligt 11:04, 2 January 2007 (UTC)
Thank you Steaphen 21:13, 4 January 2007 (UTC)

[edit] Suggestion, re mediation

I propose that further dialogue on this page (and edits to the main article) be suspended until the mediators hoe in. If they (informal mediators) don't respond, then I'll bump it to a request for formal mediation.

Cheers, Steaphen 21:01, 28 December 2006 (UTC)

  • Rather than omit a widely accepted solution/work-around from the lede, as you seem to be proposing, I have added MENTION of 2 other PROPOSED solutions --JimWae 23:18, 30 December 2006 (UTC)
  • I believe this is enough to now merit removal of the NPOV template --JimWae 06:31, 8 January 2007 (UTC)

Pending mediation, please do not remove the notice concerning the lack of neutrality in this article. Steaphen 02:57, 21 January 2007 (UTC)

  • How about a few words on just what you find to be POV in that paragraph? I long ago made amendments to that paragraph - despite your lack of clarity even before on what you objected to. What in it do you still find POV? --JimWae 04:51, 21 January 2007 (UTC)


Does anyone other than Steaphen find this article to be POV? Assuming that he is the only dissenter and everyone else agrees that its fine, I'm not sure why we need the POV warning - if every article which had a single person complaining about it had a warning, then wikipedia would be full of them. GordonRoss 05:54, 21 January 2007 (UTC)

Actually, I suppose that something like Zeno's paradox is always going to be controversial it might be a good idea to keep the POV warning on the article anyway, just to alert readers to the fact that this sort of stuff doesnt have unamimous agreement. GordonRoss 05:57, 21 January 2007 (UTC)

Response by Steaphen
I have asked for the courtesy of abiding by the mediator's direction on this issue. To suggest that one person may not raise an objection smacks of a totalitarian mindset that has become all too prevalent in our times.
If you have nothing to fear, then allow due process. I have raised a quite valid objection (as explained in the above talk section) that with the experimental evidence of quantum physics, strict determinism (by way of the use of geometric series) is not a valid treatment of Zeno's Paradoxes.
You seem to want to avoid or fear to question how an arrow, hare or runner can move through quantum scale increments ... which they MUST do (at some point) given that the accepted correlation of geometric series with physical movement requires a continuum through all scales, ad infinitum.
Your logic, or lack of, in the face of incontrovertible evidence is, I believe, a quite startling example of a disconnect between theory and observed reality.
The demonstrated obedience to accepted dogma (as shown on this issue) is, as far as I'm concerned, no different to the blind obedience to religious dogma that Galileo must have faced.
Once again, the mention of geometric series as a solution to Zeno's Paradoxes should only be done so in a historical context -- as being an outmoded conceptual framework that is no longer congruent with observed reality.
Steaphen
Author, Awkward Truths: Beyond the dogmas of science, religion and new-age philosophies
www.SteaphenPirie.com
  • So, because it does not present as THE ONLY solution the POV which you (& no other editor, and very few in the field) favor, and does not denigrate all other proposed solutions, you think it is POV. Does that sum it up? --JimWae 22:13, 21 January 2007 (UTC)


(Final) response by Steaphen (January 22, 2007)

I have previously given a link to a website Quantum Philosophy THEORIES! that is noticeably more disciplined in intellectual rigour than this site.

That site (among others) advances theories that go towards explaining the observable, verifiable facts.

From my perusal of articles, not one of them advance the nonsense that is posted at this site.

As for your reference to "THE ONLY SOLUTION" ... it seems you have not availed yourself of the rich and diverse field of quantum phiosophy, and the many -- often conflicting -- theories to explain the observed facts.

For example, here is an impromptu snapshot of that site... (taken today, January 22)

Next PAPERS online by: 2) Prof. Panu Raatikainen -Complexity and information 3)Prof.A.Khrennikov -Quantum-like formalism for cognitive measurements- 4)Prof. Gil Oliveira Neto -The de Broglie-Bohm Interpretation of Evaporating Black-Holes- 5)Dr.Owen Maroney,Prof.B.Hiley -Consistent histories and the Bohm approach. 6)Dr.David Wallace -Worlds in the Everett interpretation- 7) Prof. Tony Sudbery -Why am I me? and why is my world so classical?- 8)Dr.Gao Shan -Quantum superluminal communication must exist- 9) Prof. Yvon Gauthier -The Mathematical Foundations Of Quantum Mechanics by Hilbert and Von Neuman- 10) Prof. E.Bieberich -Structure in human consciousness: A fractal approach to the topology of the self perceiving an outer world in an inner space- 11) Prof. Brian D. Josephson - Biological Utilisation of Quantum NonLocality- 12)Prof. Ruth Kastner -Geometrical Phase Effects and Bohm’s Quantum Potential and TSQT Elements of Possibility-

The dialogue that I have experienced at this site has confirmed for me its light-weight, vapid nature.

This is my final post with this site. You may do as you wish to the main article. I only ask that you not delete, disfigure or edit my posts.

Thank you.

Steaphen Pirie Author, Awkward Truths: Beyond the dogmas of science, religion and new-age philosophies Steaphen 00:37, 22 January 2007 (UTC)


Perhaps you should find a paper on it that has _direct_ relevance to Zeno's Paradox and cite it. However given that the site you linked doesnt seem to have complete papers on it , I doubt youll find any there. (did you actually bother to read that site, or did you just type "quantum philosophy" into google and find it listed as the second link? If you want to link to a bunch of totally irrelevant quantum physics papers, then at least take the time to search through the google hits until you find something respectable, such as this) GordonRoss 00:47, 22 January 2007 (UTC)

[edit] Specific errors of the Wikipedia entry on Zeno's Paradoxes

The main article of this entry of Zeno's Paradoxes, as of January 21, 2007 follows *largely* a mathemtical solution to the issue of Zeno's Paradoxes. That is to say, the overall theme of the main article is to give some validity to the idea of geometric series as a viable solution to the issue.

For example, consider this quote (from the section "Proposed solution using mathematical series notation")

although Homer must pass through an infinite number of distance segments, most of these are infinitesimally short and the total time required is finite. So (provided it doesn't leave for 2h seconds) Homer will catch his bus.

This demonstrates the underlying misunderstanding or ignorance of quantum physics, concerning particles and the motion of particles (and collections of particles) at quantum scales, let alone any movement at smaller, unsubstantiated "infinitesimely small" scales.

To theorise concerning the movement at such scales requires clear statements of how physical movement occurs not only at "infinitesimely small" scales, but at much larger quantum scales.

No mention is given of quantum theory which is central to the issue of how physical things might move at such "infinitesimally small" scales.

The overall theme of the main article is highly biased towards outdated classical views that are no longer congruent with observed experimental evidence.

The entire page should be rewritten by, or moderated by, a reputable quantum physicist, perhaps giving passing mention to geometric series as an archaic framework that held some dubious tenure prior to quantum theory.

As far as I am concerned, the responses to my questions and posts have demonstrated a substantial lack of intellectual rigour that, if representative of other articles, demonstrates the failure of the Wikipedia model to be anything other than an adhoc blogfest.

Steaphen 08:42, 21 January 2007 (UTC) Pirie

07:53, 21 January 2007 (UTC)

You asked for "a reputable quantum physicist". I hope I count. I have PhD in theoretical physics. I'm currently a Lecturer in a major Australian university (roughly equivalent to an Assistant Professor in the US) in physics and my speciality is quantum many body theory. I think the classical view point is correct for this type of article. The main reason for this physicists tend us the phrase "Zeno paradox" to refer to the classical paradox, and say "quantum Zeno paradox" when they want to discuss the quantum case. The is already a small section on the quantum Zeno paradox - certainly this could stand to be expanded (sorry I don't have the time right now) and possibly is important enough to warrent it's own page. Therefore it seems to me that the page reflects proffesional current usage. I should add that I'm new to editting Wikipeadia - I hope this is helpful, sorry if it's not.

Thank you, ?
I came back to the site to catch the link to this page for an email I'm sending to a surgeon who's intellectually very sharp and who's “on the ball” re the interplay between quantum physics/spiritual concepts and everyday life.
I'm at a loss to understand what you mean by "professional current usage" ? professional ?? ... Zeno made some observations that deserve better treatment in light of quantum theory, irrespective of currently accepted "professional" dogma ... and I don't use the term "dogma" lightly. I've followed the travesty of Dr Ted Steele (if you're Australian, you'll know that he was hounded out of the UK for daring to question accepted Darwinian dogma, even though he had 20+ years of experimental evidence to support his theories) ... nothing short of a disgrace. “Objective scientists” ... hardly. No, when it comes to venues that demonstrate clarity of fact and statement without recourse to old outmoded frameworks of belief ... wikipedia is not one of them, at least not from my experience regarding this issue.
The good news is that I'm now in gear to develop a site that will be devoted to clarity of thought ... it will showcase articles from perceptive writers that sense or understand the inherently irreducible paradoxical nature of life. My medical friend, I hope, will be one of them (he's already contributed articles to a similar site in the U.K. It comprises scientists and medical folk focused on such matters – of how subjective/spiritual/quantum-physical phenomena can be framed within a fuller everyday-reality context).
so this whole exercise has been quite productive for me. It's highlighted the failure of the Wikipedia model ... its lack of sufficient boundaries and discipline (and thus paradoxically, sufficient freedom) to nurture the solidification of order ("fact") from chaos ("opinion") ... the paradoxical process of life.
Thanks to all. Ciao
Steaphen 01:02, 3 February 2007 (UTC)

[edit] Instant/Moment

  • instant
    • noun: a particular point in time
    • noun: a very short time (as the time it takes the eye blink or the heart to beat)
  • moment
    • noun: a particular point in time (Example: "The moment he arrived the party began")
    • noun: an indefinitely short time (Example: "Wait just a moment")

Which meaning is "meant" by "instant" or "moment" will affect the exposition. I take Zeno's instant to be without duration - especially since he claims there is NO motion at each instant --JimWae 04:40, 4 February 2007 (UTC)

[edit] Hi.

I added a tiny piece to the section about space/time not being infinitely divisible. Hope thats ok. —The preceding unsigned comment was added by 82.35.111.77 (talk • contribs) 12:52, 13 February 2007.

Someone removed my bit. Why, what was arong with it? —The preceding unsigned comment was added by 82.35.111.77 (talk • contribs) 17:30, 16 February 2007.

  • My edit summary says: "remove interesting original research that overlooks diff between physical divisibility & mathematical divisibility"
  • It was interesting and somewhat plausible, but it did overlook an important difference - and was WP:OR --JimWae 17:48, 16 February 2007 (UTC)


[edit] Hi.

as we know the closer you come to the speed of light the more time dilation comes into effect [i.e time slows down the faster you go] a particles way to move is not just throught space but by a bending of time also.

homer moves towards the bus a fractionally small amount of time dilation comes into effect and allows homer to approach the bus [the particle is moving and so is moving through time/space ,

as the bus is not moving throught time/space at this point away from homer [the particle] he is able to approach the bus. if the bus was moving away from homer at a faster rate through space/time ,homer would not beable to catch the bus. a very simplified version i know but you have to start somewhere!


and now the non-simple version...

A general geometric series can be written as


which is convergent and equal to a / (1 − x) provided that |x| < 1 (otherwise the series diverges). Again, if we attempted to solve for a/(1-1) - a/0 we are back to the problem that nothing can be divided by 0. The geometric series that would solve this to convergence reflects our perceived reality, if 0 were that point at which there were no more distance to travel and the destination had been reached. Albert Einstein had written that "what we are sure about in reality is not reflected in mathematics and what we're sure about in mathematics is not reflected in reality." This paradox on the surface is a reflection of the conundrum of math vs reality.

Although these proposed solutions effectively involve dividing up the distance to be travelled into smaller and smaller pieces, it is easier to conceive of the solution as Aristotle did, by considering the time it takes Achilles to catch up to the tortoise, and for Homer to catch the bus.

In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms-1) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms-1 with x > 1. It takes Achilles time d / xv seconds (s) to travel distance d and reach the point where the tortoise started, at which time the tortoise has travelled d / x m. It then takes further time d / x2v s for Achilles to travel this new distance d / x m, at which time the tortoise has travelled another d / x2, and so on.

Thus, the time taken for Achilles to catch up is seconds. Since this is a finite quantity, Achilles will eventually catch the tortoise.

Similarly, for the Dichotomy assume that each of Homer's steps takes a time proportional to the distance covered by that step. Suppose that it takes time h seconds for Homer to complete the last half of the distance to the bus; then it will have taken h / 2 s for him to complete the second-last step, traversing the distance between one quarter and half of the way. The third-last step, covering the distance between one eighth and one quarter of the way to the bus, will take h / 4 s, and so on. The total time taken by Homer is, summing from k = 0 for the last step,seconds.

taken from main page.

[edit] Wikipedia:Mediation_Cabal/Cases/2006-12-28_Zeno's_Paradoxes_and_Geomeric_series

It seems as though there is still some dispute over what I would characterize as a central tenet of this article. A previous mediation request unfortunately did not get off the ground due to a mediator problem, but I am willing to try to sort this out. There has already been significant discussion here, and this talk page is getting a bit long, so I would like to conduct the mediation at this page. If everyone could start by going there and responding to the questions, it would be much appreciated. Anyone and everyone is welcome to participate. --JaimeLesMaths (talk!edits) 02:07, 12 March 2007 (UTC)

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