Talk:Zeno's paradoxes/Archive 1

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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

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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)

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)

Except that mathematics crossed that hurdle, and it's the philosophers (who are NOT mathmatitions) who are arguing. Most everyone with a math PHD will tell you it's resolved. 128.255.142.190 (talk) 23:15, 5 March 2008 (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.

• • Not quiiiiite... the problem is that those of us who study math know that the specific problem is solved (It really is, we've moved so much beyond even infinities...) The problem is that philosophers don't trust stronger mathematics, they only trust weak mathamatics fields. (Like addition, and subtraction, and multiplication.... basic arithmetic/algebra, but not calculus) —Preceding unsigned comment added by 128.255.142.190 (talk) 23:19, 5 March 2008 (UTC)

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?

--

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)

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)

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.

The simple solution to the arrow paradox is derivatives. Silly. —Preceding unsigned comment added by 128.255.142.190 (talk) 23:22, 5 March 2008 (UTC)

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)

I think its also to do with continuous/discrete space-time, and so a continuous distance for example COULD be divided an infinte number of times theoretically. Even if that definition isn't concordant with whatever set-theory. —The preceding unsigned comment was added by 129.11.197.210 (talk) 20:47, 28 April 2007 (UTC).

The Universe is infinite in its total space. It goes on forever. a fraction of infinity (infinity devided by anything) can only be infinity. Any amount of space is a fraction of the total space of the Universe, which is infinite. So with that knowledge, to move through any amount of space is to move through infinitity, which would take forever, and thus could never be completed. The size of the units keep getting smaller (halving, if you will) but there is always an infinite number of them. You would have to transcend units of space, to sort of blink in and out of infinity. How can you move from one end of forever to the other? There is no other end of infinity, you'll never each it. You can't move at all. 71.59.170.19 22:09, 30 September 2007 (UTC)Daniel

The above explanation was me. I wasn't logged in. Caelub 10:27, 1 October 2007 (UTC)

The Universe is infinite in its total space. - There is no world view that claims this. Creationist views generally describe _only_ God as infinite, implying the universe is finite. The big bang is even more clear, with specific numbers proposed for the total size and mass of the universe. It is also false that any fraction of an infinite quantity must be infinite. If such a fraction is finite, then it contributes 0% of the whole, but that is the simple definition of the relationship between finite and infinite quantities, and so does not disprove anything. Algr 00:53, 12 November 2007 (UTC)

It is also false that any fraction of an infinite quantity must be infinite. If such a fraction is finite, then it contributes 0% of the whole, but that is the simple definition of the relationship between finite and infinite quantities, and so does not disprove anything.

Division of an infinite quantity is not defined in math (otherwise we have: infinity/x=0 => infinity = o*x = 0 ... so infinity = 0 ?). As pointed out here, there is confusion of the idea of infinite quantities. It is most simple to understand the paradox as Zeno allowing for infinite time and finite actions (contiguous in space). So the paradox is that time will exhaust all actions between two states, but why is it fair to _assume_ that time is infinite and actions are finite? These _assumptions_ need to be plainly stated because the results easily follow from the assumptions, hence this merely seems to be "begging the question" (since the assumptions are omitted, the reader is being deceived into believing the paradox is one of reality not the assumptions). Moreover, there is another implicit assumption occurring, and that is discrete states must be contiguous in space given time: what I mean is, a movie is a discrete set of states that appears continuous because the frames of the movie are not contiguous actions, there is a gap in time between frames. But, yes, necessarily if one has a map of actions with an infinite domain (time) and with a finite and contiguous image (actions) then the image will be exhausted due to the compactness of the number line . . . it seems to me that this is simply mental masturbation concerning the Midpoint THM and hence the fact that there are infinite real numbers between 0 and 1 (albeit _assuming_ the midpoint THM holds for our infinite time and finite actions).

Contrarily, if one has continuous infinite time and actions then the calculus of infinitesmals holds and there is no paradox. Moreover, it is sad that the calculus argument is dismissed on this page as being absurd because "obviously, one cannot take infinite action". There are far too many assumptions on this page to deserve a place in an encyclopedia (i.e. IMO the paradox arises out of our absurd assumptions). Hence, I concur, there is a "Bizarre Definition of Infinity" and more so there is little clarity in laying out _any_ of the fundamental concepts.

Hence, I would merely point out that this contradiction is a proof (by contradiction) that either 1 both time and actions are infinite (or both not) 2 with infinite time and finite actions, actions need not be physically contiguous (movie frame idea)

Yet, it would seem that this page is ambiguous about these results (or ambiguous about the validity of calculus)? Then, is a wikipedia page seriously proposing that if I shoot an arrow it doesn't move, or that I can never catch another runner? or that calculus is erroneous? - poor form wikipedia, poor form. —Preceding unsigned comment added by Sirmacbain (talk • contribs) 18:12, 9 December 2007 (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.

But one can reach the end of something infinite. Because there's a difference between infinite bounded and infinite unbounded —Preceding unsigned comment added by 128.255.187.32 (talk) 01:49, 6 March 2008 (UTC)

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)

to finish a sequence that has no finish is a logical contradiction

Syntactically yes. But conceptually, no, an infinite sequence can exists between two finite points, hence the contradiction is a confusion of words. An example is the real numbers between 0 and 1, that is (0,1), which has a cardinality (size) greater than N (the size of the natural numbers). The set of elements of an infinite sequence has cardinality N (the size of the natural numbers). So, (0,1) has a "start" and a "finish" and is a set with larger cardinality than the elements of a sequence (N<R).

This page (not this section) is confused with laymen's discussions of infinities, yet the laymen's proof of merely firing an arrow etc. is refuted? What needs to happen: the assumptions here need to be axiomatized so as not to confuse the reader, because there are significant (I would argue illegitimate) assumptions going on here that are unnoticed. And, only a deceiver should wont to guise their assumptions behind verbiage. sirmacbain 17:48, 11 De 2007 (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)

Then why are they in the external links section, instead of the references section? Homestarmy 21:13, 27 February 2007 (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.{{POV-section}}" 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?

RE: You're close to it! Look at the new edit in the section about arguments against the turtle and dichotomy paradoxes. That would be the right (and proved) explanation. —Preceding unsigned comment added by 61.2.57.254 (talk) 07:05, 6 March 2008 (UTC)

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 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.

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.

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)

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.

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)

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

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)

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)

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)

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)

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)

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)

Hi.

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

Someone removed my bit. Why, what was arong with it? —Preceding unsigned comment added by 82.35.111.77 (talk) 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)

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.

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)

Inaccuracy in the text

The article states: "...,but on the rational number line, there is for any rational number still no next rational number either." This is not entirely true. Even if the usual ordering dosn't work, there are several orderings which will give the next rational for any rational. The problem is just about how we define what that elusive next rational number means. That is not in fact a big problem at all, because it shoulnd't matter what do we call the next number. We know that the number is even if it's a bit hard to describe it. 130.234.198.85 23:55, 7 May 2007 (UTC)

There IS a smallest number

The anser to these paradoxes is that time is as a loop, a loop using the smallest number, which combines real numbers with unreal numbers. 0.0(repeating)1, i.e. 0.00000000000000...00001, with the ... repreenting an infinite number of zeros. Thus, time is not a series of "nows", rather, an infinitely huge number of infinitely small loops which compreise a larger infinitely small loop, which in turn is an infinitely small part of a bigger infinitely small loop, and so on. Thus, motion DOES continue, even in what he terms as now's (This also explains why time is slowed by motion, you cannot make a true line longer, but you can make a loop larger.) Please someone tell me if I'm crazy. --74.134.8.244 03:22, 10 May 2007 (UTC)

Nah, it's too easy :-). By the way, why exactly four zeroes after the infinitely many? Can't you get a number one tenth as big, by putting five zeroes after the dot dot dot? --Trovatore 05:39, 19 May 2007 (UTC)

• Speaking of repeating decimals, did 74.134... know that the really repeating decimal 0.99999... EXACTLY equals 1? --JimWae 05:40, 28 June 2007 (UTC)

The Arrow Paradox

If you could separate time into a series of "nows", then, that would not be motion. Therefore, you're talking about instants, and not about motion. So, the paradox is really irrelavant to motion. The moment you divide time into "nows", you are no longer talking about motion. Therefore, saying motion does not occur is false.

Feel free to refute this, I just spent like two minutes thinking about it so there are probably obvious flaws. Also send me a message.

Alex.petralia 18:33, 2 June 2007 (UTC) Alex: In fact, Zeno convinced himself there could be no motion. A E Francis 02:10, 28 June 2007 (UTC)

I think this is a good page

Despite all the wrangling on this discussion page, I think the Zeno Paradox page is a good overview. This is always a difficult topic: even Heisenberg and Einstein never came to an agreement. Perhaps it might be useful to add a section on Bertrand Russell, who attempted to solve the Zeno Paradox as it applied to the "everywhere closed packed neighborhood" mathematical space (ie. a Banach Space, or Hilbert etc.) He was unsuccessful. Also, it might be useful to talk about how Zeno's paradox led him to conclude that there could be no motion; Leucipus and Democritus answered this with the "atomus"; the particle which was ultimately small. In this way, the Greeks discovered quantum mechanics, by concluding that space could not be infinitely divided, as mass could not be inifinitely divided. It is a difficult subject. I have read many math books which never reach an agreement on the subject. Also, this led to the doctrine of the "One", which led to the determination that there had to be a God; the "unmoving mover". That may be beyond this page. Just some thoughtsA E Francis 02:10, 28 June 2007 (UTC)

A note to JimWae

Jim, I noticed you dropped the link from Zeno to the Thought of Thomas Aquinas Part I. I also noticed that you thought the article was a personal essay. Tell me how to improve it, if you will. Let me introduce myself: I am a spinal surgeon, and I have a law degree, and a master's in quantum chemistry and computational chemistry. I write advisory reports for US federal judges, who hire me, personally. To be honest, I don't care whether the link to the Zeno page is there, or not. It would appear the woman who put the "essay" tag on my article doesn't really know much about the subject. Anyway, if we want to make this better, tell me how to improve it.A E Francis 02:45, 3 July 2007 (UTC)

Clean up?

Just a note: I ran across this page while looking up other related articles. I found the subject very interesting but the way that the information/arguments are presented somewhat put me off of reading it. I believe the sections regarding Proposed solutions would be better off in a seperate section from 'issues with proposed solutions'. Also, there is almost a lack of cohesion as it progresses. It has the feel of a document that was dictated by several parties at the same time. As evidenced by the lengthy talk page, it is obvious there is considerable debate about the content. Perhaps a third party could help clean up the main article so it would be easier to read and more understandable. Just an idea. ProGeek314 20:46, 28 August 2007 (UTC)

Different orders of infinity

There is a section in the article beginning with the claim that some people say that Zeno's paradox has something to do with infinite sets having different "orders", by which is meant having different cardinalities. The section refutes this connection so completely that one is left wondering if anyone really says this. I think a reference should be provided. If this is a common error made with respect to Zeno's paradoxes, then, by all means, this section should stay. However, at the moment, I think that it should be removed. —Preceding unsigned comment added by Clifton Ealy (talk • contribs) 15:30, 8 September 2007 (UTC)

"Logically impossible"

What confuses me the most about the "Issues with the calculus solution" section are statements like this:

"...for Achilles to capture the tortoise an infinite series of physical events need to be completed, which is logically impossible."

Why is it logically impossible to perform an infinite number of events? I see no reason at all that this should be true. If this is the crux of the paradox, then Zeno has created a paradox by introducing a false axiom. Maybe we should be honest and say something to the effect of:

"This paradox assumes the following axiom: 'An infinite number of events cannot be completed.' "

Almost right: the axiom would be: "An infinite number of events cannot be completed sequentially". And the reason why this would be an axiom, is that if you try and go through all elements of an infinite set one by one, you'll never reach the end. Is that so hard to see? I really don't understand why you would claim that there is "no reason at all that this should be true". Yes, fine, so calculus shows that the *sum* of an infinite number of values can be finite, I have no problem with that. But the paradox here is not about the sum of any infinite series: it's the *process* of going through such an infinite series that's the whole quandry. Now look, if you're telling me that there is no actual such infinite series of actual physical events in the actual physical world, and that the 'events' we are referring to do not have actual physical counterpoints, well, that's ok, and that would indeed be a solution to this paradox. But it wouldn't be *calculus* then that would solve the problem! And by the way, all of this is carefully laid out in the article. —Preceding unsigned comment added by 72.226.66.230 (talk) 16:24, 19 October 2007 (UTC)

See my comments below. Sthinks 04:01, 1 November 2007 (UTC)

To me, talking about "events" is just a bunch of conceptual labeling. I can label the events however I want to. Zeno decided to label an infinite number of events which clearly happen in a finite amount of time. The fact that they happen in a finite amount of time is what makes it "possible" for the sequence of events to be completed. Sthinks 04:29, 24 September 2007 (UTC)

But if this is all just about conceptual labeling, please note that the resolution of the paradox would then be just that: semantics. Not calculus. —Preceding unsigned comment added by 72.226.66.230 (talk) 03:49, 25 September 2007 (UTC)

Well, it depends on the way that you frame the question.

This article favors a version of the paradox that goes something like this: "A real arrow must complete an infinite number of events in order to travel a finite distance. But an infinite number of events cannot be completed. Since real arrows actually do travel finite distances, there must be a paradox!" I think this version of the paradox is a silly semantic trap because there is no ground for the second sentence. As I asked above, why can't an infinite number of events be completed? There is no reason why not.

One could also phrase the paradox in the following manner: "A real arrow must complete an infinite number of events in order to travel a finite distance. How is it that an infinite number of events can happen in real life? That seems strange!" In this version of the paradox, calculus is very useful. It shows that, no matter how we assign arbitrary labels to the "events" in the question, we always see the same trajectory of the arrow. In other words, calculus helps us show that the underlying physical process makes mathematical sense, that an infinite number of events can be completed in a finite time, and that the labeling of events is arbitrary from the point of view of physics. (Whether we divide the incident into one event, two events, or an infinite number, in any of the infinite ways we can do so, we always get the same answer: the trajectory of the arrow is x(t)=vt, and the total time to hit a target at distance d is t=d/v.)

Personally, I would like to see a comment like this incorporated into the article. But I'll leave it to the discretion of others. Sthinks 22:27, 27 September 2007 (UTC)

Sthinks, you seem to be overlooking the point of the paradox. Calculus assumes that position and time are continuous variables. For the purpose of this paradox, Zeno accepts the same assumption. Say an object travels 1 m/s. The mathematics tells us that at the end of 1 s the object will have travelled 1 m. It also allows you to calculate the position of the object at any instant in time in between. All of this is uncontroversial and Zeno would agree. But Zeno now poses the difficulty of how an object can ever move from point A to B, on the assumption that space is infinitely divisible, and that there are an infinite number of points between A and B, all of which must be occupied along the way. It does not solve the problem to say that time is also infinitely divisible, and to map each of the inifinite number of instants with an infinite number of corresponding positions. In the spirit of the paradox, Zeno would reply that the same problem arises with regard to time. The problem then becomes how can time ever progress from instant A to instant B, there being an infinite number of instants which must occur in between? The math shows that the object will have travelled 1 m at the end of 1 s, and .5 m at the end of .5 s, and so on. This is all well and good. But the math can no more show that 1 m will ever be covered than it can show that 1 s of time will ever in fact elapse. Even if it were assumed that an infinite series of "events" could take place in a finite period of time, how is it clear that any finite period of time will ever elapse, given that an infinite number of instants must occur between any two other instants?

In other words, you cannot solve the problem of whether one infinite series (position) will ever be exhausted by positing a second infinite series (time) and saying that both series will be exhausted together. That assumes exactly what was required to be proved: namely, that an infinite series is in fact exhaustible. —Preceding unsigned comment added by 24.126.153.193 (talk) 15:58, 28 October 2007 (UTC)

I feel like I've already rebutted the argument that you gave, but let me try to be clearer. I assert the following: the labeling of events is arbitrary and has nothing to do with reality. If you add the conceptual label "infinite" to some process, you don't change the process. If you confuse yourself in the process, it's your own fault.

Now, it's an interesting idea that the paradox might boil down to the question: "How can time happen at all?" I think this is a much deeper question than the paradox as stated. One might ask: Why is it that we move through time at a "finite rate" (whatever that means)? Special relativity provides a little bit of an answer to this question, but not much. The reason we believe in time is because we experience it consciously (and it is an excellent tool for making scientific predictions). So there are plenty of valid questions here, but the paradox doesn't illuminate them. From the perspective of physics, it remains an empirical fact that we do seem to move at a "finite rate" through time and, if we assume this, it solves the paradox.

So why did I bring up time? Well, the primary assumption of the paradox seems to be, "An infinite number of events cannot happen." This seems patently absurd to me, since an infinite number of events are always happening. An infinite number of events happen during every single instant. But people seem to believe this statement, and they often invoke time in many implicit ways when they bring it up. For example, a fellow who commented above said that "An infinite number of events cannot happen *sequentially*." The word "sequentially" is a reference to time, so time is inherenty part of the problem. (S)he also described the process of "going through the members of an infinite set one by one." This is another reference to time, describing a temporal or psychological process. From a mathematical perspective, we don't have to "go through" the events. They are all already there -- pure mathematics doesn't have a concept of time. From a physical perspective, "going through the events one by one" means watching them and seeing how long it takes to finish each one. In this perspecive, calculus solves the problem.

Contrary to his/her claim, none of this is really addressed in the article. The assumption there is presented in a very vague and dogmatic way: "...an infinite series of physical events need to be completed, which is logically impossible." From each of the perspectives I have given, this is not logically impossible.

Now I'm beginning to think that taking more math classes would resolve a lot of people's problems with this paradox. If you take any math past calculus, it becomes quite clear that a continuum like the real numbers is defined in a completely different way (and has different properties) than any "discretuum." The paradox may boil down to a misunderstanding of those properties. To me, from a math perspective or a physics perspective, none of this is disturbing at all. Time is just like space, and I can label an infinite number of points on a continuous curve through spacetime, but that does not necessarily mean that an infinite number of "events" are happening, and even if it did this shouldn't surprise us. All we did was say, "Look! Here are an infinite number of points!" There is no reason why there cannot be an infinite number of points on a curve, nor any reason why an infinite number of events cannot happen. Sthinks 03:49, 1 November 2007 (UTC)

In reply to Sthinks,

That "an infinite number of events cannot happen" it not a necessary assumption. The Dichotomy paradox suggests the following problem: that in progressing along a continuum, since there is no next point to reach, it is not possible to reach any different points at all. Start at a given point. In order to reach any different point, "other" different points must have already been reached (and proceeded through), namely the points between the given point and the different point. Thus no different point can ever be reached, unless a different point already has been reached. But how can a different point be reached before any different point is reached? That is logically impossible. Put a different way: suppose we are to start at 0 and must build out the real number line towards 1. Since we cannot reach any number greater than 0 unless we have already reached a number greater than 0, how will we ever reach any number greater than 0? You have to do something before you do it. Now if we are constructing an abstract mathematical number line it's fine to say that the points are all there in one go because we put them there with one conceptual act. And if we are to post hoc divide up something that has already occured, then of course as an abstract matter we can divide it up in any way that is logically consistent (whether it corresponds to empirical reality or not). But it is something different if we are supposing that there is an actually existing empirical spatial continuum that is isomorphic to the real number line, and that an object is actually trying to move across it. In that case, these points are not there because we put them there in our minds (such that we can take them away again whenever we please, or just choose to look at the problem in another way), these points are empirically real.

The line of reasoning which says that "labeling of events is arbitrary and has nothing to do with reality" is perfectly true as far as it goes, but Zeno is questioning the nature of the continuum itself (and is assuming for the purpose of the paradoxes that such a spatial continuum empirically exists). Do you therefore believe that there are no such empirical continua? Then the Achilles and Dichotomy paradoxes are indeed resolved (though Zeno has other paradoxes based on the assumption of discreteness). Or do you believe that space (for example) does consist of an empirical continuum of points? Then we are right back to square one. Or do you actually mean, when you say that "labeling of events is arbitrary," that space and time are not empirically real at all and have no essential nature of their own? Or that somehow time and space are neither continuous nor discrete? Under this category of answers there are so many different beliefs and responses possible (some of which could be deep and meaningful) that it would be useless to hazard guesses as to your true meaning. You would have to enlighten us. In any case, the labeling of events, while arbitrary, has no bearing on what space and time are in themselves.

To return briefly to the argument to which I had responded initially. The whole point of these two paradoxes is to question how progression across a continuum is possible. As far as any purely calculus based solution is concerned, the space and time continua are represented by two independent continuous variables, each of which have the same properties. How can it be any more believable, then, that an object can progress through a temporal continuum, than it can be believable that it can progress through a spatial continuum? If someone questions how the latter is possible, they will certainly not accept the former either, for they come to the same thing. The continua are identical. All that mapping individual points to individual instants can ever prove is that if a certain finite time interval has elapsed, then a corresponding spatial interval has also elapsed. But the fact that a finite time interval ever will elapse is not a priori any more believable than the fact that a finite spatial interval can elapse. As far as calculus is concerned the continua are identical. This "solution" entails assuming exactly what the paradox is questioning. How is this mathematical handwaiving anything more than begging the question?

The statement that "From the perspective of physics, it remains an empirical fact that we do seem to move at a "finite rate" through time and, if we assume this, it solves the paradox," is once again, missing the point. Obviously it is a matter of empirical fact that time progresses. But to say that that "solves" the paradox is silly. It is also empirically the case that objects move across space. Why not just go ahead and say that the empirical fact that objects move across space "solves" the paradox? Now, if objects didn't move across space or time, then there would be no paradox (because there would no longer be any possible contradiction between Zeno's arguments and reality). That motion across continua is an empirical fact is the only reason there is a paradox. I would not call that a "solution" of the paradox.

 —Preceding unsigned comment added by 24.126.153.193 (talk) 19:50, 10 November 2007 (UTC) 

This is Sthinks writing again:

Here is my solution to the "dichotomy paradox." For t=0, the particle is at x(0)=0. For t>0, the particle is at some point x(t)>0. There is no paradox. Once again, you're trying to treat a continuum like a discretuum. The solution is: "At some particular instant in time, the particle leaves x=0. For all later points in time, it is at a point x(t)>0." Now let me go into more detail.

The "dichotomy paradox" is a kind of a reductio ad absurdum argument, and I like it better than the "infinite number of sequential events" kind of argument that was given above. It seems reasonable on first glance. But you seem to be expressing one of two things, and one of them is the argument I've already discussed. (You say: "Before the particle could move to x=.1, it had to move to x=.05." I say: "Indeed, it did." You say: "And before that it had to move to x=.025." I say: "Of course." You say: "But then it had to go through an infinite number of points." I say: "Yes, and it did all of that in a finite amount of time, so it's physically possible for that to happen.")

The other argument, which is very interesting, goes like this. We repeat the discussion above for a while, and at some point you get frustrated at this point and say: "But when was the first moment that it wasn't at x=0? (When I phrase it this way, it's called the 'dichotomy paradox.')" And I say: "Aha! This is the fatal assumption you've been making all along. In a continuum, there is no first moment. There is no well-ordering principle on the continuum. There is merely an infimum of all of the moments that it was not at x=0, but no minimum, just as there is an infimimum on the set (0,1), but no minimum. The continuum doesn't behave like a finite set, and you're expecting it to, and that's why you came up with a paradox." And hopefully that resolves the argument.

Here is another way to express this. You say, as above: "But how can an object be at a different point 'before' ever being at any different point?" I say: "Let's make this more precise. You're saying that we should choose a particular point a>0. To get to a, the particle has to pass through a/2. This is true. But you think this shows that the particle cannot move from '0' to 'non-0' instantaneously. This is false. The dilemma just arose because you chose a particular point in your 'proof,' rather than considering the set of all points {x|x>0}." A topologist might resolve the dilemma by saying: "Consider three sets: A={0}, B={a}, and C={x|x>0}. What you actually proved is that A and B are not connected sets. But you claimed to have shown that A and C are not connected sets. The first statement is true but the second statement is false. A and C really are connected."

This may seem like a strange mathematical cop-out, but it isn't. It's very intuitive. You can see with your own eyes that '0' is right next to 'non-0,' but x=0 is not 'right next to' x=a. One could express this precisely by saying that the infimum of the distances between points in A and B is a, but the infimum of the distances between points in A and C is 0.

Here are two other comments:

-I believe that space and time are real continua, though (as I wrote below) no one can verify this experimentally. My comments about "conceptual labeling" were intended to show that it shouldn't matter whether we decide to say "the particle moved from 0 to 1" is a single event or an infinite number of events (0 to .5, .5 to .75, .75 to .875, etc.). The underlying physical process is the same.

-I think that my comments on the "physics perspective" are being taken out of context. Everyone other than you has implicitly used time in their "proofs" of the paradox. (For example, the word "sequentially" is a dead giveaway.) Since they were already assuming that time exists, I went ahead and used it. Your formulation is the first one that I have seen which does not refer to time. Sthinks 04:21, 12 November 2007 (UTC)

After I came home, I had one more idea. As you can see from above, my basic objection to the "dichotomy paradox" is that it seems like you are looking for the first moment when the particle is not at x=0, when no such thing is well-defined. (Nor need be -- there is no reason why there should be a first moment. A similar situation exists in discrete cases. For example, what is the smallest member of the set {1/n|n in N}?)

You probably think this is a misrepresentation of your argument. If so, I challenge you to come up with a formal, axiomatic argument that does not implicitly use this assumption. I would love for you to prove me wrong by coming up with an excellent argument.

For your first two axioms, I would suggest:

1) A particle's path in space is described by a continuous path x(t): R -> R.

2) If x is a continuous path and x(0)=0 and x(b)=a, then there exists a point t in (0,b) such that x(t)=a/2. (This is a special case of the Intermediate Value Theorem in math, but we'll just take it as an axiom here.)

How you continue this list of axioms is up to you. I hope you like this challenge. I think this is much more exciting than most of the other discussion that has happened here. Good luck-- Sthinks 07:43, 12 November 2007 (UTC)

I'm disappointed that no one has taken up my challenge so far. Anyway, if someone did, here is where I would expect the flaw to be:

In the argument for the "dichotomy paradox" given above, someone wrote that "we cannot reach any number greater than 0 unless we have already reached a number greater than 0." Because of sloppy language, (s)he basically interpreted this to mean, "We cannot reach any element of the set {x|x>0} without first passing through a point which is a finite distance from 0." This sounds kind of similar to the axiom, but it is false. The clear, correct version of the axiom is what I called "axiom 2" above. You cannot derive the dichotomy paradox using this axiom. (I would love for you to try, though.)

To me, the moral of the story is this: if you prove a theorem which is clearly false, then you started out with a false axiom. This is especially true of a subject like mathematics where the axioms are purely formal. If the axioms give us results that give us an absurd idea of a continuum, we can always choose different axioms that give us a sensible idea of a continuum. And we did develop a sensible idea of the continuum in the 19th century, but a lot of philosophers refuse to study it.

My challenge is still open for anyone who wants to undertake it. Good luck! Sthinks (talk) 06:15, 25 November 2007 (UTC)

Continuity in Quantum Mechanics

I am a physics Ph.D. student, so I think I have an obligation to weigh in on the quantum mechanics debate. I've read about half of the posts on the subject so far and all I've found is a mess of unclear thinking and strange associations. (I would have read them all, but I couldn't take it any more.)

1) Every generally-accepted physical theory is formulated in the context of continuous spacetime. This includes nonrelativistic ("normal") quantum mechanics, all quantum field theories, and even string theory (which is highly speculative anyway). In quantum mechanics, we say that the "position operator" has a "continuous spectrum" of eigenvalues. This is just a fancy way of saying that there is a continuous spectrum of possible answers to the question, "Where is this particle?" I should emphasize that the word "continuous" is the operative word here; spacetime really is assumed to be continuous.

On the other hand, many physical systems are "discrete." The most famous example is the energy levels of an atom. A bound electron in an atom really must "jump" from one discrete energy level to another. Mathematically, this is because the "Hamiltonian operator" of a simple atom has a "discrete spectrum of eigenvalues." The main point here is that some quantum systems are discrete, but not all of them are. In particular, spacetime is always assumed to be continuous, and no evidence has been found to resolve the question one way or another. (Continuity is assumed because the theory is more elegant and easier to use that way.)

2) There are a few theories (but only a few) which have attempted to discretize space and time. All of them are unpopular because discretizing spacetime presents enormous difficulties mathematically and gives few practical results. The only theory which gets much attention at all is Loop Gravity, which proposes to unify GR and QM by discretizing spacetime. So far, most physicists remain unconvinced of the value of this approach. (On a technical note, to prevent confusion, I should say that Lattice Field Theory regularly approximates spacetime as being discrete. But even in LFT it is assumed that the underlying spacetime is really continuous and that the lattice approach is merely an approximation.)

I hope this helps resolve the dispute here. Sthinks 04:30, 24 September 2007 (UTC)

It appears to me that you have confused or equated possibility (as may be framed by a quantum wave function) with actuality (as observed, e.g. via photon detectors, or with our eyes: otherwise known as the collapse of the continuous wave-function).

If continuity of space was an actuality, then we could use geometric series to plot or predict the behaviour of the stuff which makes up "space." Instead, we use mathematics, which in assuming continuity, cannot then 'track' the particle. Why is that? Obviously because the mathematics do not entirely apply only to this probability, but others as well (as per Dr Deutsch's arguments, among others -- namely, multiverse theories). The continuity would be of the multiverse, not necessary of our particular uni-verse.

If you have any mathematics that perfectly predicts continuous behaviour of matter in our space-time, then please enlighten me and the scientific fraternity.

Quanta (of matter and energy) do not move perfectly continuously, but jump from location to location, without moving through the space in between. Ipso facto, our particular space-time (which is composed of quantum stuff) likewise cannot be perfectly continuous.

Put differently, existing mathematical tools do not precisely predict (match) observed reality -- either those tools are "blunt instruments" that need sharpening (or replacement), or they are precisely revealing the deeper non-physical (multiverse) nature of reality.

Again, this issue is quite simple. Geometric series (as is generally used to resolve Zeno's Paradoxes) cannot be used on quantum scales. I would think that the intransigence on this issue would be quite similar to what Galileo must have faced with those priests who refused to look through the telescope.

The more interesting dialogue is about what that in-between "stuff" is made of, given that it nonlocally non-physical.

Steaphen 18:13, 27 October 2007 (UTC) -- just a brief visit to this site. Not likely to return anytime soon (the arguments in reply predictably avoid dealing with the central issue)

Steaphen: Your entire post is basically meaningless. I could go through it point by point, but my rebuttal would go something like this: geometric series are entirely appropriate in plenty of quantum contexts, but they are completely irrelevant to the question of whether or not spacetime is continuous; you're clearly misinterpreting Deutsch's work on Everett-like multiverses; quantum field theory gives exact predictions for all known phenomena in a continuous spacetime; and so forth. But your post itself shows that you really don't know what you're talking about. I'd be happy to discuss this further, but only if you can give me a clear and coherent argument using the actual physics that you're discussing. Sthinks 03:25, 1 November 2007 (UTC)

As I explained elsewhere (in previous posts to this site), the use of geometric series to solve Zeno's paradoxes assumes a 1:1 correspondence (between physical points in space, and the points along some equation -- e.g. a parabola that describes the flight of an arrow -- I refer here to the theory of limits, as used to 'solve' the paradoxes). As such, geometric series are inappropriate to resolve the paradoxes. I notice your vague and evasive responses such as "basically meaningless" and "I could go through it point by point" .. well then, do so.

Once again, please enlighten me how you can apply a point-by-point geometric progression to the movement of quantum particles. And if you assert that arrows and the like classical objects do not move at some stage through quantum scale increments, then please explain how those particles (that collectively form such things as arrows, hares the like) move .. through? what?.

Regarding your "quantum field theory gives exact predictions for all known phenomena in a continuous spacetime" Really? Exact predictions. It seems I must be a silly-billy, accepting Heisenberg's Uncertainty Principle, which limits the predictability of physical energy/matter. I must assume you have mathematics to transcend or eclipse his basic principles of not being able to perfectly define physical attribute-conjugates such as momentum and position, energy and time etc.

Btw, I find it interesting that you sign your name via some obscure handle. If you have confidence in your theories, sign your real name. No need to sneak around, offering indirect responses as you have.

Steaphen 09:14, 3 November 2007 (UTC)Steaphen

Steaphen: Seriously, you don't understand what you're talking about. Even your mention of "possibility" versus "actuality" shows that you've misunderstood the whole point of quantum mechanics. Bell's theorem (and Aspect's experiments) showed that there is no local realist interpretation of quantum mechanics, which is the assumption you seem to be making.

Anyway, if you insist on geometric series as the central issue here, then here is a way that you can "apply a point-by-point geometric progression to movement of quantum particles." You can write the state of any particle in its position representation: |psi>=int(<x|psi>|x> dx). Assume |psi> is some wavepacket that describes a localized particle. Then apply the geometric argument to each of the position-components of this wavepacket.

My comment about "exact predictions" in QFT is correct as it stands. For example, QFT can predict the Lamb shift exactly. (That's an energy splitting between two energy levels of Hydrogen.) You seem to be interpreting "exact predictions" as "exact predictions of every observable quantity simultaneously," which cannot be done by any theory. But this is a ridiculous way to interpret "exact predictions." The point is that if I decide to measure a momentum, for example, then QFT will give me all the information I can know about that momentum. And so far, no experiment has contradicted these predictions.

But this is all irrelevant. The point that I originally objected to is that you said repeatedly that space is discrete. I wrote this post to state quite clearly that space is assumed to be continuous in quantum mechanics. That is NOT the same as saying that quantum particles behave like classical particles, which seems to be how you're interpreting what I said. Quantum particles don't behave even remotely like classical particles (or even superpositions of classical particles). But that has nothing to do with whether or not space is continuous.

As for my name: that's a cheap ad hominem, and it has nothing to do with the correctness of either of our arguments. Also: I don't see your last name listed anywhere. Are you hiding something? (No, you aren't -- it's just that people use anonymous handles on the internet, and it's a standard thing to do.) Sthinks 03:54, 12 November 2007 (UTC)

Thank you for your "flat-earth theory" response ...'it looks flat, so it must be ..." quite entertaining, but tiresome.

Regarding your "you seem to be interpreting "exact predictions" as "exact predictions of every observable quantity simultaneously," which cannot be done by any theory.

I would have thought you would at least attempt to be a 'good scientist' by matching observable reality with theory. To suggest close enough is good enough is rather sloppy thinking, I would have thought.

Again, it is quite simple. See my earlier post, re points A and B, the trajectory of an arrow, and using standard Newtonian physics to predict all its properties, simultaneously. Standard Newtonian physics most definitely expects simultaneous knowledge and prediction of all quantities -- that's classical physics, e.g. momentum and position of an arrow as it travels through the air. That again reminds me, where's my Nobel Prize?

As for the cheap ad hominem ... haven't got Google open at the moment, so not sure what that means.

I must away .. my apologies, but your similar responses to others has confirmed my energies are best directed elsewhere.

Kind regards,

Steaphen Pirie

Author, Awkward Truths

ps. I suggest you be alert to your use of such phrases as 'always' 'nowhere' and other such absolutes. Your use of such terms telegraphs your dogmatic beliefs ... to which of course you are welcome, but others will be alerted to your mindset.

Steaphen: I'm not going to respond to your trash-talking, but there are two things worth saying:

-If you still believe in local realist physics, as your comments on "all observables simultaneously" seem to indicate, you should really learn more about Bell's theorem and the EPR experiments by Aspect, Weihs, and others. If you truly understand what these experiments show, it will blow your mind. Nowadays, one can gather enough data in five minutes to show, by 100 standard deviations, that local realism is not possibly true. It's true to a very good approximation for macroscopic objects, but cannot be ultimately true for all objects.

-I'm not really interested in continuing this conversation right now, but I am interested in figuring out why you think what you do. So I tried to find your book. So far as I can tell, it doesn't exist. Who published it? Where can I get a copy? Thanks-- Sthinks 04:47, 13 November 2007 (UTC)

My figuring, as you put it, is based on observed reality, particularly in the field of quantum physics. More is explained here. Steaphen (talk) 22:34, 30 April 2008 (UTC)

Size Paradoxes

I noticed that although there is mention of his motion paradoxes and indeed the very basis of the size paradoxes (infinitely divisible) there is no discussion (unless the math thing is a discussion of it, my math ended with calculus) of his size paradox. I.e all things are either size 0 or infinitely large b/c they can be infinitely divided and the sum of all parts is the whole ( how big are the pieces? how many are there?). This is a response to the pluralists. I.e 0+0+0+0=0, Aristotle proposes the solution that the actual size of the whole is divorced from the parts, and that the parts exist in potentiality (same with the atomists). Furthermore there is > a countably infinitely # of parts in a continuum aleph1, and because of this adding the parts does not yield the sum but only the countably infinite amount of aleph 0. If this is already in the text forgive me, if not I'll gladly glitz it up and source it (kirk raven scofield the presocratic philosophers edition 2). moomoo2u69.157.240.152 21:38, 8 November 2007 (UTC)

I think its called the extant antimenes or something like that. moomoo2u —Preceding unsigned comment added by 69.157.240.152 (talk) 19:45, 13 November 2007 (UTC)

Chinese Paradoxes

I'm going to delete that two similar chinese paradoxes. They really aren't particularly well phrased or developed, and their mention is somewhat out of place with the rest of the article.(Lucas(CA) (talk) 06:12, 16 December 2007 (UTC))

Moved from article to talk page for discussion before entry

These entries are totally WP:OR and there are attempts to present the editors conclusions as definitive conclusions, and also unencyclopedic uses of editor's voice to even say "this is one editor's opinion". An encyclopedia is not an editorial page. -- --JimWae (talk) 06:03, 19 December 2007 (UTC)

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bY User:Liongold (User talk:Liongold Special:Contributions/Liongold)

It is possible to solve the arrow paradox if we consider that the arrow, even while in 'rest' during an instant, is still moving because of the energy it has. Therefore, even though it has a position, that position can never be exact because of its velocity, which makes it move even when it is in Zeno's instant of time. It is somewhat like the uncertainty principle; we cannot know both its speed and its position, even though the arrow makes use of both so that, while we can be clear on one term, we remain uncertain about the other. In this way, we can see quite clearly that Zeno's arrow paradox can be solved with the uncertainty principle of quantum mechanics.

Also, Zeno did not include the notion of speed in his paradoxes, which can easily defeat both the dichotomy and the Achilles paradoxes. If we consider that Achilles is moving faster than the tortoise, then that will allow him to cover the distance faster. The faster he runs, the closer he gets to the tortoise, until he finally reaches the position of the tortoise and runs faster than the distance the tortoise can make, provided the tortoise has a slower speed than Achilles. If he continues to move faster than the tortoise, he can easily race past the tortoise and get to the end of the race. In this way, the Achilles paradox can be solved.

The dichotomy paradox requires not speed, but rather the input of energy to solve it. The energy of a wheel in motion will allow the wheel to move infinitely fast when we consider fractioned distances e.g. 1/16 of a distance to reach 1/8 and so on.The energy lost will be very small as the distance is extremely small. This process continues as the distances get larger and the speed slower of the moving wheel, until it stops because it no longer has enough energy to move fast enough to cover the distance required to make another rotation. In this way, so can the dichotomy paradox be solved.

These are only suggestions, however, and are as yet unconsidered by anyone apart from the editor of this page.

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by: Unknown

May I note, that the use of 'motion' by Zeno and probably all of western society that days, was a totally other notion than that of ours. While we can speak of a relative motion, the Greek had only an absolute on. This is also seen by the problems between Galileo and the Church, who still used the Ptolemaic/Aristotelian system in which the earth was the center of everything. In his famous arguments, Galileo points out that motion is relative instead of absolute. For that he was put to death, but we must now say he was right. What arises here is that the problems of Zeno's paradoxes are not our problems. The problem with the notion of 'motion' is not our problem cause we have another notion of 'motion'. For more information about the big changes in the use of words by the enlightenment, you could search for 'mathesis universalis': this was the 'moment' where the difference between heavenly knowledge (absolute, eternal, unchangeable) and earthly knowledge (always in change) collapsed, and Descartes proposed the method of seeing everything through the mathesis universalis: everything is calculable by one single method of knowing. This also made possible the new natural studies like 'chemistry', physics, etc... who then used kind of the same method based on the mathesis universalis.

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I'd just like to say that the first comment above is a blatant misuse of physics. We're considering a kinematic problem that has nothing to do with energy. Also, the problem was phrased in terms of "classical-like" (not quantum) physics, and it can be solved by classical physics without any reference to the uncertainty principle. All that is necessary is to use a sensible idea of the continuum in mathematics. (See the "Logically Impossible" post above.)

The second comment is largely correct but mostly irrelevant to the discussion. Zeno was concerned with how any kind of motion can happen at all, and the question of whether we consider absolute or relative motion is not important to the solution.

Original Research and babble

Quite a few sections in this article are most unsourced and babble on about about things like Planck time and the Planck length. An anonymous blog post is used a source. This article sounds like someone copied an essay for Intro to Philsophy 101. Therefore, I have added "original research" claims. Secondly, gratuitous use of "I" and "we" betray the non-encyclopedia tone of the article. Third, few footnotes are used, since the article references history and philosphy, which are prone to original research, this is especially egregious. The section on planck time and length does not say anything important at all. A major rewrite of this article is needed.Thegeneralguy (talk) 15:34, 19 April 2008 (UTC)

Dichotomy Paradox

Zeno, in the dichotomy paradox, is not saying there is NO distance between Homer and the bus (nor does he hold that the distance is infinitely great). Neither does he state that we cannot KNOW what the distance is. A mathematical solution that derives the sum of an infinite series thus, while giving a method for DETERMINING the distance, does not settle the question of whether one can TRAVEL the distance. Mathematical solutions do not disprove the paradoxes, nor do the paradoxes disprove any mathematical solution. Paradoxes are apparently true statements that appear to contradict other apparently true statements or intuitions. The very existence of a paradox cannot be disproven, though we do attempt to solve them by examining the premises of the arguments/statements involved. --JimWae (talk) 21:31, 4 April 2008 (UTC)

Agree. Paul August ☎ 22:34, 25 April 2008 (UTC)

spelling?

I always thought that it was xenon? after a search of google I found xenon zenon and zeno are any of these alternative spellings?

Time is NOT "irrelevant" to the paradox.

Over and over, this article keeps asserting false axioms as if they were obvious. As with most paradoxes, this one is founded on an unreasonable implicit assumption. Several people have pointed this out, but the passages which point out the implicit assumptions are always removed from the article. Specifically, I object to the following passage:

"The following thought experiment can be used to further illustrate the fact that any quantification of time is irrelevant to the paradox. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle at all, then apparently Achilles must have stopped counting, and we could ask Achilles what the greatest number he counted to was. But of course this is nonsense: there is no greatest number, and Achilles can never stop counting. Therefore, Achilles cannot catch up with the Tortoise, whether he has finite time or infinite time to do so."

The false assumption here is: "If he catches up with the turtle at all, then apparently Achilles must have stopped counting." The only reason that anyone believes this axiom is because no human being can count to infinity, because it takes infinitely long to do so. We believe this line of argumentation because of our everyday experience with time. But there is absolutely no reason why Nature cannot count to infinity. In physics, she does it all the time. (Notice that, as usual, the implicit assumption is introduced with the phrase "of course.")

Put very simply, this argument seriously confuses two things:
-The inability of a human observer to count an infinite number of "events."
-The impossibility of an infinite number of "events" occurring.
These two things are completely different, and yet the article treats them as if they were the same.

As I have said before, even if we label an infinite number of events in some process, the same physical process can happen. Nature does not care how we label them. The entire "paradox" is a semantic trap. We label physical processes with an infinite number of "events" and then bewail the fact that we have found a way to label more events than we are able to count out loud. None of this has anything at all to do with what is actually happening physically.

All versions of Zeno's paradoxes--including the so-called "dichotomy paradox"--rely on some false axiom (usually related to time) that is only easy to believe because it agrees with our everyday experience, not because it must be true. This is why no one has answered my challenge to prove Zeno's paradoxes axiomatically. (I made this challenge above in the section called "Logically impossible.") To me, this is evidence that the paradox cannot be proven. Sthinks (talk) 19:01, 3 May 2008 (UTC)

The article needs to be completely rewritten by someone who is familiar with the academic literature on this topic.

In practical terms, I think that the entire article needs to be scrapped. Someone should find articles from all the major philosophers (not just Kevin Brown) who have debated the paradox and summarize their viewpoints. But I suppose that we should not try to "prove" or "explain" the paradox in this article, because none of us can agree on its validity or even how to present it.

I'm interested in hearing what others have to say about this proposal. Sthinks (talk) 19:06, 3 May 2008 (UTC)

• I do not recall seeing much problem with the presentation of Zeno's points. Where the article goes wrong is in the repeated and *lengthy* presentations of opposing arguments and answers to opposing arguments. Short statements of arguments & references to further discussion elsewhere could still work, I think --JimWae (talk) 21:46, 3 May 2008 (UTC)

Someone with more time write more to the article about this

http://plato.stanford.edu/entries/paradox-zeno/#Sta http://plato.stanford.edu/entries/paradox-zeno/#ArgDen http://plato.stanford.edu/entries/paradox-zeno/#ArgFinSiz http://plato.stanford.edu/entries/paradox-zeno/#ArgComDiv —Preceding unsigned comment added by Le Blue Dude (talk • contribs) 16:51, 4 May 2008 (UTC)

Advice

Quit warring over contect: Quit POVING. Quit letting the opinion of the few influence the opinion of the many. If necessary start a separate page for justifying why Zeno's paradox is still/is no longer a paradox. with a title of, for example: Zeno's paradox solution. Keep it to the known facts, please. —Preceding unsigned comment added by 64.6.88.31 (talk) 15:48, 4 May 2008 (UTC)

Actually, screw it, carrying out my own advice. If I can improve Wikipedia, I should. Le Blue Dude (talk) 16:10, 4 May 2008 (UTC)

Proposed philosophical solution to Zeno's paradox

I would like to present to your judgment a philosophical solution that I am proposing for Zeno's Achilles and Tortoise paradox

I will start with a statement so obvious it’s almost always left unmentioned. A description of any phenomenon is not identical to the phenomenon itself.
In this case the phenomenon consists of two objects, Achilles and Tortoise, and process of motion of these objects. Descriptions of this phenomenon... well they can be different, and you are free to pick the one you like. For example, you can pick a description featuring infinite but converging series of distances or times. You can also choose a description based on physical parameter “average speed” and write a simple equation where average speed multiplied by time gives you distance. Fantasy permitting, you can even construct a description of your own to account for this famous race*. You are also free to choose Zeno’s description, of cause.
The important thing to realize is that whatever description you are using, if it provides you with conclusions (about when, where and whether Achilles is going to catch up with the Tortoise and any other conclusions your description might provide), these conclusions can be verified by observation and proved either correct, satisfactory or incorrect, unsatisfactory. Accordingly, you can judge particular description as being adequate or inadequate based on quality of conclusions it provides. However, quality of description in no way affects the described phenomenon itself.
Let’s return to Zeno’s description. What he outlines for us is an infinite process of dividing distances/time into smaller and smaller parts. This process can not be completed. That’s the bad news, of cause. But the good news is that it is the process of description of the phenomenon that can not be completed, and this problem in no way affects the process of the phenomenon itself i.e. the process of Achilles’ and Tortoise’s movement. Zeno’s description goes awry - bad for Zeno, but why should Achilles care?
Would Zeno realize that description of any phenomenon is not identical to the phenomenon itself he should have finished his speech with different words. Instead of “therefore, swift Achilles can never overtake the tortoise” he should have said: “Therefore, I am unable to finish my description of the race and come to any kind of conclusion about where, when and whether Achilles is going to overtake the Tortoise”. No paradox, just a bad way to describe the phenomenon.

• an alternative description of Achilles vs. Tortoise race, aka “trjam-pam-patic dives theory”

Achilles being a fast runner, fabled hero and all graciously allows the Tortoise a head start of a hundred feet. Little does Achilles know that the Tortoise, while a slow runner, is a one of a kind turtle-hero himself and is capable of trjam-pam-patic dives that occur as follows. Every time someone comes within one foot of the Tortoise it performs an instant leap forward through the space-time continuum, a dive that guarantees that the Tortoise appears 3 feet in front of its chaser. Obviously, whenever Achilles comes within one foot of the Tortoise it will trjam-pam-patic dive 3 feet ahead and swift Achilles will never overtake the Tortoise.

“This description is ridiculous”, you might say. I hope so. I actually tried to come up with something funny. The “trjam-pam-patic dive theory” is based on a rather arbitrary and loosely defined concept that has no supporting evidence whatsoever. More important, this “theory” leads to conclusions that can be verified by observation and proved wrong. Just like Zeno’s description.
Edit
It appears that statement "A description of any phenomenon is not identical to the phenomenon itself" can be misinterpreted leading to some confusion. Thus, user 128.113.89.96 attempts to dismiss the argument by saying "Semantical solutions to the paradox claim that the mathematical description of the situation is not mirrored by actual physical events. However, as long as there is a physical point corresponding to the mathematical halfway point..." etc.. By doing so he completely misses the whole point (pun intended). It is not the actual physical events that have to mirror mathematical or any other kind of description. It is the description, whatever it is based on (be that algebraic concept of real numbers, geometrical concept of dimensionless "points", concept of "quantum of space" or your grandmother's will) has to try and mirror actual events as they observed in the real world. Whenever description fails to produce adequate predictions, it should be abandoned and a better description should be searched for. And this is exactly what happened with Zeno's description - work of Isaac Newton and Gottfried Leibniz produced mathematical methods of handling infinite sequences and adequate descriptions of the Achilles vs Tortoise race have been found.--Orlangoor (talk) 04:03, 22 April 2008 (UTC)

I agree with Orlangoor. Zeno's description of the situation, which assumes that between any two points in space (or time) there is always another point, leads to the prediction that motion is impossible. But clearly there is motion, and so Zeno's description needs to be abandoned. —Preceding unsigned comment added by 72.226.66.230 (talk) 13:03, 25 April 2008 (UTC)

I already solved Zeno better under "Bizarre Definition of Infinity" below. Basically, all of Zeno's paradoxes are based on making a finite task seem infinite by dividing it an infinite number of times. But the division doesn't reflect the true nature of the task itself, so the task itself isn't infinite - just Zeno's measurement (or description) of the task. Algr (talk) 07:33, 1 May 2008 (UTC)

How is this solution 'better', Algr? Seems to be exactly the same as what is pointed out at various places, including in the article itself. Moreover, in your "Bizarre Definition of Infinity" entry you still refer to calculus as a proper solution, but that's very strange, as calculus does assume that any finite amount can be divided into infinitely many pieces, which is exactly what leads to all these problems, as you and many others have pointed out. Therefore, calculus doesn't solve the paradox at all. Unless you get to terms with that, you in fact haven't provided any solution at all. —Preceding unsigned comment added by 72.226.66.230 (talk) 13:29, 1 May 2008 (UTC)

Annon, sign your posts with four "~", and use ::: to indent your replies. My solution is better because it describes all the paradoxes, not just Achilles and Tortoise, and it doesn't get lost in irrelevant details like "trjam-pam-patic" which describes a totally different race for no reason. Also note that I wrote it a year and a half ago. The core of Orlangoor's solution seems to be the same as I wrote anyway. I don't see anything like it in the article itself, another poster commented on that. I referred to calculus as a result of my solution, not as something backing it up - the logical core of my solution is what I wrote above, and does not involve calculus. Algr (talk) 17:09, 1 May 2008 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Zeno's paradoxes/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Comment(s)

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Geometry guy 23:14, 9 June 2007 (UTC) The math here is severely confused in how it is being treated outside of the context of the presentation of calculus. There seems to be significant confusion by most authors about the infinite - which does significant harm to the sequence analogy. Hence, there needs to be a transition from foundations (i.e. cardinality etc.) to dispel the confusion. Moreover, the assumptions presented in the paradox need to be enumerated to appropriately address to topic. Sirmacbain (talk) 19:01, 9 December 2007 (UTC) The mathematics here is irrelevant if it is not congruent with the experimental evidence (of how Zeno's arrow, runner and hare move). There appears to be a frightful disconnect of many mathematicians with reality (as exemplified by this rating). The mathematical arguments presented here reveal a deep misunderstanding of the scientific method. Mathematics is one tool (there are others) by which we seek to explain and predict physical reality. If that explanation and prediction fails, then the tool needs to be sharpened or replaced. As a direct result of this basic misunderstanding of the scientific method, as exemplified above and on the talk page, formal mediation has been requested to correct the glaring disconnect and incongruency between theory and evidence. Steaphen (talk) 19:32, 18 November 2009 (UTC)

Last edited at 19:32, 18 November 2009 (UTC). Substituted at 21:03, 4 May 2016 (UTC)