Talk:Zeno's paradoxes

Bibliography

This list is intended to collect references thought to be relevant for the article. Delete entries only when they are blatantly and obviously inappropriate. In general, we want not only to collect useful references, but also be able to check new additions against previous discussions that lead to exclusion. Provide diffs, and update section links when they get archived.

The 2001 edition of Salmon's anthology lists at least 218 sources, so it is safe to say that this bibliography cannot be considered anywhere near comprehensive before we have passed the 200 mark.

• Grünbaum, Adolf (1967). Modern science and Zeno's paradoxes. Wesleyan University Press. http://books.google.com/books?id=eSpDAAAAIAAJ. Retrieved 13 February 2010. 

• Grünbaum, Adolf (1968). Modern science and Zeno's paradox.. http://books.google.com/books?id=kakEPAAACAAJ. Retrieved 13 February 2010. 

• Salmon, Wesley C. (March 2001). Zeno's paradoxes. Hackett Publishing. ISBN 9780872205604. http://books.google.com/books?id=0AzP9WLLJLcC. Retrieved 13 February 2010. 

Salmon's book is one of the best on the subject. Huggett, in his article "Zeno's Paradoxes" in the Stanford Encyclopedia of Philosophy [1] writes: After the relevant entries in this encyclopedia, the place to begin any further investigation is Salmon (2001), which contains some of the most important articles on Zeno up to 1970, and an impressively comprehensive bibliography of works in English in the Twentieth Century . Paul August ☎ 14:22, 13 February 2010 (UTC)

The bibliography of my 1970 hardcover edition has 143 entries, the 2001 edition cited above has at least 218 (preview limit, sorry). Paradoctor (talk) 08:32, 25 February 2010 (UTC)

• Alper, Joseph S.; Bridger, Mark (1997). "Mathematics, Models and Zeno's Paradoxes". Synthese 110 (1): 143–166. doi:10.1023/A:1004967023017. ISSN 00397857. 

Abstract from the official page at Springer: "A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time."

• Sewell, Kip K. (1 October 1999). The Cosmic Sphere. Nova Publishers. ISBN 9781560726616. http://books.google.com/books?id=X24ifBRL_7kC&pg=PA410. Retrieved 1 March 2010. 

Pages 14-15 (section 3 "Infinite Time" of chapter 1 "the Container of All Things") discuss the arrow paradox.

Footnote 10 on page 410 (for page 15 in section 3 "Infinite Time" of chapter 1 "the Container of All Things") discusses "proposals at the ability to cross an infinite provided infinite acceleration is assumed".

From Amazon's author page (WebCite): 'Kip Sewell holds an MLIS from the University of South Carolina and currently works as an information professional. He has also received BA and MA degrees in Philosophy and has been a college lecturer. "The Cosmic Sphere" (1999) is Sewell's first work on the subject of cosmology. He is currently revising the book and continues to explore issues in science, philosophy, and theology as an independent researcher.'

Apart from this book, Scirus, Google Scholar and WorldCat turned up nothing by Sewell.

IMO, a minor primary source, apparently not peer-reviewed, by a philosopher very early in his career. Paradoctor (talk) 01:17, 2 March 2010 (UTC)

• Paul A. Fishwick, ed. (1 June 2007). "15.6 "Pathological Behavior Classes" in chapter 15 "Hybrid Dynamic Systems: Modeling and Execution" by Pieter J. Mosterman, The Mathworks, Inc.". Handbook of dynamic system modeling. Chapman & Hall/CRC Computer and Information Science (hardcover ed.). Boca Raton, Florida, USA: CRC Press. pp. 15–22 to 15–23. ISBN 9781584885658. http://books.google.com/books?id=cM-eFv1m3BoC&pg=SA15-PA22. Retrieved 5 March 2010. 

Defines "Zeno behavior", a concept from the field of verification and design of timed event and hybrid systems.

• Verelst, Karin. "Zeno’s Paradoxes. A Cardinal Problem - I. On Zenonian Plurality" (PDF). PhilPapers. http://philsci-archive.pitt.edu/archive/00002731/01/RIGAproceedings.pdf. Retrieved 5 March 2010.  arXiv:math/0604639

Criticizes the "Received View" on Zeno as untenable. Maintains that a "generally overlooked" key to Zeno arguments is that "they do not presuppose space, neither time". Paradoctor (talk) 17:33, 5 March 2010 (UTC)

Her official homepage (5 March 2010) Paradoctor (talk) 19:52, 5 March 2010 (UTC)

to do

• Paul Hornschemeier's most recent graphic novel, The Three Paradoxes, contains a comic version of Zeno presenting his three paradoxes to his fellow philosophers.
• Zadie Smith references Zeno's arrow paradox, and, more briefly, Zeno's Achilles and tortoise paradox, at the end of Chapter 17 in her novel White Teeth.
• Brian Massumi shoots Zeno's "philosophical arrow" in the opening chapter of Parables for the Virtual: Movement, Affect, Sensation.
• Philip K. Dick's short science-fiction story "The Indefatigable Frog" concerns an experiment to determine whether a frog which continually leaps half the distance to the top of a well will ever be able to get out of the well.
• Allama Iqbal's book The Reconstruction of Religious Thought in Islam discusses the paradox in Lecture II The Philosophical Test of the Revelations of Religious Experience, and suggests that motion is not continuous but discrete.
• Ursula K. Le Guin's character of Shevek in The Dispossessed discusses the arrow paradox in great amusement with his un-understanding classmates as a child.
• add missing refs from Rucker section below
• add refs deleted with this edit

Classic troll

Zeno appears to be the classical equivalent of the Forum Troll. 202.74.196.251 (talk) 01:30, 12 October 2010 (UTC)

This comment was probably intended to be humorous, but it might be closer to the truth than many of the assertions within the article. Take a look at http://www.mathpages.com/rr/s3-07/3-07.htm, where Kevin Brown, one of the philosophers opposing the view that Zeno's paradox is solved (or even can be solved) mathematically, reviews the history of the problem. Lapasotka (talk) 11:17, 12 January 2011 (UTC)

You spend ages waiting for a bus, and then none come along, because they haven't been invented yet.

The Dichotomy problem obviously wasn't originally expressed in terms of Homer trying to catch a bus. Would it not be better to express it in whatever terms Zeno actually used? Wardog (talk) 15:05, 13 January 2011 (UTC)

It would. Ansgarf (talk) 01:34, 24 January 2011 (UTC)

None of Zeno's writings have survived. What we have of the dichotomy paradox comes by way of Aristotle and is quoted in the article: "that which is in locomotion must arrive at the half-way stage before it arrives at the goal." Paul August ☎ 01:56, 24 January 2011 (UTC)

Removed from article until improved

Archimedes, the conventional solution, and infinite processes

The solution proposed by Archimedes is a proper mathematical treatment of Aristotle's notion that the time it takes to cover the increasingly smaller distances is reduced likewise. Since the parardox is not explicit about the rate of speed at which Achilles catches the tortoise, or how far away he is, we are free to assume that Achilles is catching the tortoise at a constant rate of 1 metre per second, and that he is 1 metre behind the tortoise. Then the time taken to cover each distance, as per Zeno, can be modeled as a sequence and the infinite sum of this sequence is 1/2+1/4+1/8+..., which is equal to 1 (see geometric series for a proof of this fact). According to this model, we can calculate that it will take Achilles exactly 1 second to catch the tortoise.

A stipulation that Achilles is gaining on the tortoise at a constant speed (as a function of time), or something similar, is necessary. After all, if Achilles isn't travelling faster than the tortoise, he isn't going to catch it. In fact, only the closing speed needs to be known - the absolute speeds of Achilles and the tortoise are irrelevant to the paradox. The tortoise may be stationary as Achilles runs towards it, or it may be that Achilles is stationary at the bottom of a slippery slope, while the tortoise slides helplessly backwards down the slope, towards Achilles.

The closing speed is constant in Zeno's paradox, where it takes Achilles 1/2 a second to gain 1/2 a metre on the tortoise, 1/4 sec to gain another 1/4 metre, 1/8 sec to gain another 1/8 metre, etc. If Zeno had used a different situation with a different series, for example, where it takes Achilles 1/2 a second to gain 1/2 a metre on the tortoise, 1/3 sec to gain another 1/3 metre, 1/4 sec to gain another 1/4 metre, etc., then Achilles is never going to catch the tortoise. He will always be catching the tortoise, but at an ever slower rate, and will get within billionths and trillionths of a centimetre of the tortoise, but will never actually catch it. The subtle difference between these two situations reflects the subtleties inherent in the mathematics of infinite processes.

It may be objected that in this situation, the motion of Achilles towards the tortoise is not only not constant, but is not necessarily even continuous (as a function of time). However, it is possible to achieve the same results with the closing speed of Achilles modelled by a continuous function - see the digamma function. What this shows is that by itself (assuming the conventional approach/model used here) knowing that Achilles is always moving forward relative to the tortoise (i.e. catching the tortoise whilst behind, travelling faster whilst level, or pulling away if he is ahead) does not entail that Achilles will ever catch the tortoise, but that we need to know how fast he is catching the tortoise. If he is catching it fast enough, he will catch up eventually. If he is catching up too slowly, he never will catch up, ever. According to Archimedes and Aristotle, then, if f(t) (where f(t)>0) is the function modelling Achilles closing speed on the tortoise as a function of time, the apparent paradox results from either ignoring this function altogether, or assuming that knowing the precise values it takes is of no relevance to the problem.

---

removed from article until improved --JimWae (talk) 20:36, 4 March 2011 (UTC)

Discussion of above

You raise 5 objections.

(1)What the hell are you talking about? Of course we can assume that Achilles is catching the tortoise at constant speed. If your aim is to disprove the statement "Achilles cannot catch the tortoise (no matter his speed)", you are free to assume anything you like about Zeno's speed. Are you stupid?

(2)This is not dubious. This is what the whole point of the section is to make clear - that a stipulation that Achilles is catching the tortoise at a constant rate, or some similar stupulation, is necessary for Achilles to catch the tortoise. I guess you could include proofs, or a link to the harmonic series, for example.

(3)Covered by (1).

(4)Is pretty obvious.

(5)This is simply related to the divergence of the harmonic function. All it needs is a link. The divergence of the harmonic function is not "dubious", having first being proved in the 14th century. Do try and keep up.

Raiden10 (talk) 22:21, 5 March 2011 (UTC)

---

• Your responses are abusive and indicate you are unfamiliar with philosophical argument. As I said, there IS a proper mathematical solution for the sum of a convergent series, even if it is not a full solution to the paradoxes. However, your wording of even the mathematical solution is unnecessarily presumptive.

• IF you assume a catching rate of 1 m/s from 1 m back, one needs only simple algebra (and does not need to know anything at all about the mathematics of convergent geometric series) to calculate that catch-up happens in one second
• ASSUMING a catching rate is begging the question, which is only ever provisionally allowed; it does not itself provide a solution, just a different approach. Zeno "is not explicit about the rate of speed at which Achilles catches the tortoise" (as you say) because HE contends catching the tortoise is an illusion -- and at least it is the point at issue. We are free to assume any distance. We are not restricted to constant speeds, but we can provisionally assume the speeds ARE constant & see what happens -- but we are not free to assume Achilles catches the tortoise, thus we are NOT free to assume a "catching rate" (constant or not) solves the paradox

• IF it takes 1/2 s to close 1/2 a metre gap, the closing rate is 0.5m/0.5s = 1 m/s.

  IF it takes 1/3 s to close 1/3 a metre gap, the closing rate is 0.333m/0.333s = 1 m/s.

  IF it takes 1/4 s to close 1/4 a metre gap, the closing rate is 0.25m/0.25s = 1 m/s.

  IF it takes 1/5 s to close 1/5 a metre gap, the closing rate is 0.20m/0.20s = 1 m/s.

• I think what you want to discuss is it taking 1 s to catch up 1/2 m, another second to catch up 1/3m, another s to catch up 1/4 m, etc. In that case, if the numbers chosen were as YOU have given them (with a 1 metre head start and a closing speed of 1 m/s -- [contrary to the rest of the article, btw ] ), Achilles catches the tortoise in less than 3 seconds. In case you still have overlooked it: 1/2 + 1/3 + 1/4 > 1. Furthermore, in this series, it takes less than 11 seconds to close a 2 metre gap, less than 31 s to close a 3 metre gap, less than 83 s for 4 m, and less than 227 seconds for a 5 metre gap. A 6 m gap would take just over 10 minutes.
• There are specific problems where a harmonic series produces paradoxical results -- BUT this is not one of them. Zeno never *gives* any numbers (except 1/2) AND discussion of harmonic series is irrelevant to this article -- besides being WRONG here. The paradoxes given in that [ie. the harmonic series ] article indicate the counter-intuitive result is that the task IS actually completed in finite time. So how is what you say about this in any way correct?

• Your re-insertion of virtually the same material (which is quite the same as was removed months ago by others) before engaging in dialog here does not indicate any attempt to work together. I encourage others to participate in this discussion. I have taken an extraordinary amount of time to discuss this with you and I have not called you stupid. The material you have added has many errors & irrelevancies. You have provided no sources, and the parts that are not wrong or irrelevant are redundant. Your contribution needs to be removed from the article AGAIN. --JimWae (talk) 00:41, 6 March 2011 (UTC)

My reading of Zeno is that in his view, his paradox was not to call attention to the fact that "the task IS actually completed in finite time", but rather to show that motion is an illusion.

As Douglas Adams put it, "Time is an illusion, tea time doubly so."

This is a subject on which mathematicians and philosophers often disagree. The article can only report that disagreement, citing an objective source. Rick Norwood (talk) 12:50, 6 March 2011 (UTC)

Rick: Note that I have clarified above that I was referring to the harmonic series --JimWae (talk) 22:58, 6 March 2011 (UTC)

---

":*

• IF it takes 1/2 s to close 1/2 a metre gap, the closing rate is 0.5m/0.5s = 1 m/s.

  IF it takes 1/3 s to close 1/3 a metre gap, the closing rate is 0.333m/0.333s = 1 m/s.

  IF it takes 1/4 s to close 1/4 a metre gap, the closing rate is 0.25m/0.25s = 1 m/s.

  IF it takes 1/5 s to close 1/5 a metre gap, the closing rate is 0.20m/0.20s = 1 m/s."

Yes, tragically I wrote 1/2m, 1/3m, 1/4m,..., where I meant 1/2m, 1/4m, 1/8m,....

Tragically???? This series is NOT even relevant to this article. How come I had to explain the very same thing twice (about ALL those rates being 1 m/s) for you to catch on to this? --JimWae (talk) 22:58, 6 March 2011 (UTC)

---

":* ASSUMING a catching rate is begging the question, which is only ever provisionally allowed; it does not itself provide a solution, just a different approach. Zeno "is not explicit about the rate of speed at which Achilles catches the tortoise" (as you say) because HE contends catching the tortoise is an illusion -- and at least it is the point at issue. We are free to assume any distance. We are not restricted to constant speeds, but we can provisionally assume the speeds ARE constant & see what happens -- but we are not free to assume Achilles catches the tortoise, thus we are NOT free to assume a "catching rate" (constant or not) solves the paradox"

That's just stupid. Zeno's paradox might as well then be the statement that "Achilles must always catch the turtle, even if he does not want to". For how would you describe Achilles NOT catching the turtle? You would do it with rates of speed.

It's not clear exactly for WHAT reason rates of speed are not allowed, other than because you don't want them to be. Raiden10 (talk) 21:53, 6 March 2011 (UTC)

Assuming a constant catching rate assumes Achilles catches the tortoise. Assuming Achilles catches the tortoise is not a valid solution. While true that that series (the one you have changed it to, which I will refer to as the "modified harmonic series") the sum of the gains in distance at no time reaches 1 metre, such is not found in the article on harmonic series, NOR is it relevant to Zeno. Your 4 paragraphs are still unsourced original research with multiple flaws, numerous irrelevancies, and redundancies that tiny tightenings will not improve. Nowhere else in the article is there such a long string of text with no sources whatsoever. Your contribution needs massive improvement & sourcing soon, or it needs to be removed. --JimWae (talk) 22:40, 6 March 2011 (UTC)

Well, what IS Zeno's argument exactly? Have you seen it?

After all, all one has to do is to show that Zeno's argument (whatever that is) does not imply that Achilles cannot catch the tortoise. Let's say we forget about whether Achilles does catch the tortoise, and simply concentrate on showing that Zeno's argument (whatever that is) does not imply that Achilles cannot catch the tortoise.

You also read here and there that Zeno in fact intended his paradoxes to prove that space and time are not both continuous. Although quantum mechanics and whatnot today predicts that they are not continuous, what I have seen of Zeno's argument does show any such thing. Anyway, the whole "motion possible implies space and time not continuous" thing is simply the contrapositive of the whole "space and time continuous implies motion not possible", so the arguments for these statements also coincide.

I feel like however one portrays Zeno's argument one will be accused of doing a "strawman", whilst Zeno's "actual argument" remains conveniently elusive.

There is a particularly inebriating argument, of course, that is common.

"One common reply is that Zeno has misunderstood the nature of infinity. Modern mathematics, it is said, has shown that the infinite sequences that Zeno generates do have a finite sum. In particular, to take the Racecourse example, the sequence 1/2 + 1/4 + 1/8 + 1/16 + . . . is equal to 1.

This reply, however, misunderstands what modern mathematics has shown. Mathematicians do use sequences such as 1/2 + 1/4 + 1/8 + 1/16 + . . . but they say that they have a limit of 1, or tend to 1. That is, we can get nearer and nearer towards 1 by adding on more and more members of the sequence, but not actually arrive at 1 - this would be impossible because we are considering an infinite sequence. So far from providing an argument against Zeno, mathematics is actually agreeing with him!" -- Francis Moorcroft, reference 5 on this lovely little page on wikipedia

The confusion here is when we take the sequence of subdivisions into which space (measured in metres, algebraically "m"), or time (measured in seconds, algebraically "t"), is divided, we can then talk about the nth term of the sequence. This is called the index. In the sequence 1/2, 1/4, 1/8, 1/16, 1/32,... the index of the term 1/4 is 2, for example. The term (1/2)^n has index n. But n is a bound variable, it has no physical interpretation whatsoever, let alone as an index of time!

This line of thought, this inebriating confusion, leads to paradox by means of this fundamental error. The indexing variable of the sequence, n, is bound by the variable binding operator "Σ".

I don't think I really have to ask, but when it comes to references, will anything do? Raiden10 (talk) 00:21, 11 March 2011 (UTC)

1. To defeat/"solve" a paradox, one must defeat the entire argument, not just the conclusion. To show Z is wrong, one has to do more than show Achilles CAN catch the tortoise -- that can be done without any math at all, by running the race. One must show what Z's argument is either unsound or begs the question
2. No source has Zeno saying the sum is infinite--JimWae (talk) 02:59, 11 March 2011 (UTC)

1. I agree, that's exactly what I said. That's exactly what I was doing. I was pointing out the very common and annoying error.

2. No source has me saying that Zeno said the sum is infinite.

1 again. The error in the argument by Francis Moorcroft that I pointed out above stems from the fact that the index variable of the sequence introduced by Zeno (by splitting the distance by halving it, halving it again, ...) is a bound variable. It only serves to range over the elements of the sequence. It has no units. It has no physical meaning whatsoever. The faulty argument above by Moorcroft derives a paradox by treating the index as if it did have a unit, the unit being time. Raiden10 (talk) 14:31, 11 March 2011 (UTC)

But this is the article about Zeno's paradoxes, not an article about Moorcroft - nor any other commentator on Z. --JimWae (talk) 19:39, 11 March 2011 (UTC)

And what I wrote wasn't really about Moorcroft - it was about the argument he gave. I showed the error in Moorcroft's argument above, but what exactly is Zeno's argument? There needs to be an argument to refute and it's not clear to me exactly what it is. Raiden10 (talk) 14:35, 13 March 2011 (UTC)

But we are not here to write our own disproofs of anyone's arguments. We need to have reliable sources for counterarguments too, see WP:OR. --JimWae (talk) 20:23, 13 March 2011 (UTC)

"Your" section is unsourced, repetitive, in some places irrelevant, and now even more repetitive. I do not see anywhere that you have made any case to keep any of it.--JimWae (talk) 20:27, 13 March 2011 (UTC)

Be that as it may, could you please answer the question? Raiden10 (talk) 03:04, 14 March 2011 (UTC)

You must be aware that we have only secondary fragments of Zeno's arguments - so nobody can say "exactly" what Zeno's argument was - we ALL have to rely on what appears in the literature.--JimWae (talk) 06:44, 14 March 2011 (UTC)

Oh, so it's not really a paradox then, is it?

If it's not Zeno's argument, then whose is it? What is the argument? No argument, no paradox. Raiden10 (talk) 04:21, 15 March 2011 (UTC)

For example, what would be your best guess at Zeno's argument? Like I said, for it to even be a paradox there has to be an argument. If people think it is a paradox, there must be an argument knocking around somewhere.

The brief argument on this wikipedia page is hardly explicit. Under the headling "Achilles and the tortoise", it starts with saying that Achilles travels at constant speed, makes several valid observations, drags on a bit, writes 5 or 6 agreeable sentences. But then suddenly blurts it out one sentence right at the end, which is not a valid deduction. It blurts out "Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise" It points two two sources, neither of which elaborate on this sentence. Raiden10 (talk) 03:10, 17 March 2011 (UTC)

Can any part of this contribution become part of article?

It seems that the best take-away from the "modified harmonic scenario"

1/2 s for 1/2 m gain,

1/3 s for next 1/4 m gain,

1/4 s for next 1/8 m gain,

1/5 s for next 1/16 m gain, etc.

is that while both the distances & times get smaller, the distances converge while the times do not. Thus Aristotle's solution "as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small" is not a "proper mathematical solution". Something like this could become part of the article, but not under the proposed solutions section. It could be given as an example of why mathematicians thought more than Archimedes & Aristotle's "solutions" was necessary (but I think the example ends up in the footnote). --JimWae (talk) 23:47, 6 March 2011 (UTC)

It's not the "modified harmonic scenario", it's just the "harmonic scenario". It's only modified from a typo. Raiden10 (talk) 23:20, 10 March 2011 (UTC)

---

About exat meaning of paradox

The dichotomy paradox “That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ” —Aristotle, Physics VI:9, 239b10

The important point of this paradox is that every locomotion must occur in the world of thought ,not real world. So, a real thing(ex:arrow,ball,bottle,apple,pen,ring.. etc) can't substitute for "that which is in locomotion". Zenon's demand is metaphysical solution.

Another important point is that nobody knows the distance between "that which is in locomotion" and the goal.

Keypoint of solution

The half-way stage that zenon said is not the point of obligation, but logical[physical] necessity. So,"that which is in locomotion" need not to arrive at the infinite half-way stages. If it arrives at the goal,it should have passed through the every half-way stages.

"That which is in locomotion" has only to arrive at the goal.

Definition

Necessity - An event that can be completed by itself and unavoidable circumstances.

Obligation - An event that can be completed by artificial locomotion and avoidable circumstances.

Reference

If you can read korean language, chek this, http://www.joubert.pe.kr/zeroboard/zboard.php?id=kisul&no=13 — Preceding unsigned comment added by Hesun (talk • contribs) 03:18, 6 June 2011 (UTC)

---

— Preceding unsigned comment added by Hesun (talk • contribs) 03:08, 6 June 2011 (UTC)

achilles and the tortoise

I don't understand why Achilles can't win the race. Is a head start necessary for this paradox? The explanation given in the article as I understand it goes like this: The tortoise starts at X + 100m and Achilles starts at X. when Achilles reaches X + 100m the tortoise is at X + 110m. So Achilles would have to keep getting slower, and slower than the Tortoise to not pass him, in which case he would eventually match the tortoise in speed, no longer being the faster runner. Also, the quote from Aristotle seems circular. 71.194.44.209 (talk) 05:09, 19 May 2011 (UTC)

Agree here - maybe I am just to uneducated to get it, but the example in the way it is written right now does not add value to my thinking. It just seems like faulty logic. Sure, after he reaches the point where the tortoise was, the tortoise moved on - just a bit - but then at some point he has simply overtaken her. So it only works for the first points, with a specific window where the tortoise has not been reached. If this "paradox" is used to explain something, then I don't see the connection. No points from me for Mr. A. 89.0.48.23 (talk) 07:48, 20 May 2011 (UTC)

Even the educated know it's faulty logic, but the trick is it's troll logic. It takes a lot of advanced mathematics to show where the error is in Xeno's thinking, just like the "Pi is 4" one where you take a square, and remove the corners repeatedly by subtracting smaller squares, keeping the same circumference until it approximates a circle. Or anything that leads to 'force-less' levitation. 75.175.216.159 (talk) 14:55, 3 August 2011 (UTC)

I too agree with the first two posters on this topic (71.194.44.209 & 89.0.48.23), and the third poster's (75.175.216.159) comment doesn't make sense to me. What does the poster mean by "troll logic?" Doesn't seem like it would require that advanced of mathematics (the example seems to come straight out of a grade school pre-algebra word problem).

The example, the way it is written (which is that both Achilles and Tortoise maintain constant speed), just seems so easily refutable - unless it is based off of some other presumption. Simply put, if Achilles is travelling faster - in the example 10x faster than tortoise - then he will eventually meet up with the tortoise:

ie if d=dist of tortoise, v=velocity of tortoise, t=interval of time, and dA is dist of Achilles,

  then d = v * t + 100, and dA = 10 * v * t.

Set both equal to each other and solve for t:

  v * t + 100 = 10 * v * t
  100 = 10 * v * t - v * t
  100 = 9 * v * t
  100/(9 * v) = t

therefore Achilles will have gone the same distance as the tortoise after 100m divided by 9 times the velocity of the tortoise (t= ~11/v so if the tortoise is going at 11m/s then it would take 1 second for Achilles to have ran the same distance as tortoise). This will be true for any velocity (cept zero of course) the tortoise will be going at given that there is no acceleration from either tortoise or Achilles.

Did read somewhere else in the article that you're not supposed to use math, or that this paradox can't simply be disproved with math... how is that? A simpler example: Achilles is twice as fast as the tortoise which travels 1 meter per hour (Tortoise's velocity is 1m/hr & Achilles's velocity is 2m/hr). Tortoise starts out 1 meter ahead of Achilles. After one hour Achilles has caught up to the tortoise.

Took some philosophy, but was a math major... maybe been too long out of school? Can one of you philosophy experts explain why this is a paradox? 166.250.0.104 (talk) 07:23, 23 January 2012 (UTC)

Ultimately, Zeno is arguing that contained within our idea of motion there is a contradiction. Paradoxes exist because there appear to be valid arguments that lead to contradictory conclusions. You have presented a familiar mathematical argument to show Achilles catches the tortoise. There is also the "common sense" "argument" in which people actually seem to catch tortoises. Zeno does not accept that there is any motion and presents an argument that appears to show that assuming there is motion leads to the conclusion that motion cannot happen. --JimWae (talk) 10:35, 23 January 2012 (UTC)

Planck length

It seems odd that there should be no mention of Planck length (a mere 16 halvings of a meter, hardly infinite) or Planck time as refutation of Zeno's paradoxes… --Belg4mit (talk) 15:49, 29 October 2011 (UTC)

Zeno paradox is a philosophical, mathematical thought experiment, not a physics experiThat ments. The fact that the paradox is resolved if you cannot divide time and space indefinitely, however is mentioned. — Preceding unsigned comment added by Ansgarf (talk • contribs) 00:54, 25 November 2011 (UTC)

That is an artificial distinction, any and all evidence which can be brought to bare should be. I do not see the text which you claim addresses this issue, so I have added a line about Planck in the modren times section. There may be a better placement, but an explicit reference should definitely be included. --Belg4mit (talk) 16:52, 26 November 2011 (UTC)

It is not an artificial distinction. Zeno's argument relies on mathematical properties of rational numbers, not on physical properties. In Zeno argument the tortoise is a point, has no weight, or length. To mention the Plank length in the paragraph on mathematics is odd, since it is not a mathematical constant. It has no bearing on the mathematical argument.

The point that you are trying to make is already addressed in the paragraph "Another proposed solution ..." in section "Proposed Solution".

Furthermore, your edit suggests that the Planck time and length are the smallest time and length possible. This is wrong, they might be the smallest time and length that you might be able to measure, but there is no consensus whether times and lengths smaller than these exist or not.

You phrased it rather carefully, and of all the people who have confused physical measurement with mathematical properties, you prosoal is probably least misplaced, and it hopefully satisfies those who mistaken think that Zeno's paradox is a physics experiment. This is just a category B article, after all. 22:13, 26 November 2011 (UTC) — Preceding unsigned comment added by Ansgarf (talk • contribs)

I have moved the contribution to the relevant section. It is also "worth noting" here that whether it is about space or about measurement of space is contested--JimWae (talk) 23:20, 26 November 2011 (UTC)