Talk:Zeno's paradoxes: Difference between revisions

Old,unsectioned comments

Removed from the main page:

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

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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.

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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".

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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

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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)

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)

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.

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)

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.

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.

"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?

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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)

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)

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)

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)

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)

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)

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)

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)

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.

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)

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)

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)

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)

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.

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)

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.

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)

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)

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)

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

• Solution to Zeno's First Paradox
• Solution to Zeno's Second Paradox (Achilles and the Tortoise)
• Zeno's Paradoxes: A Timely Solution by Peter Lynds

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)

"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 ---

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)

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)

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)

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

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.

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.)

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)

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)

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)

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)

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)

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."

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)

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)

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.

" was removed to allay concerns of a POV being espoused.

218.214.81.20 06:37, 7 November 2006 (UTC)

Some ideas about the paradoxes

Jake95 18:53, 30 November 2006 (UTC)

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.

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?

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.

--

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)

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 spacial 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.

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

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.

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.

In the meantime, it would be curteous 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 the physical (quantum) facts should be actioned as a matter of priority, lest Wikipedia gain a reputation for silly and unqualified conjecture.