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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,[clarification needed] 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.[dubious – discuss] 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,[dubious – discuss] 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.[clarification needed] If Zeno had used a different situation with a different series,[relevant?] 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.[dubious – discuss] 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)

Change to lead

I think an improvement in the lead would be to change the word "illusion" in regards to the perception of time or motion, to epiphenomenal, or Epiphenomenalism given that illusion implies a distortion or trick to the senses, whereas the perception of time is of course very normal and commonly experienced. Epiphenomenal explains that it is not a trick, but rather philosophically questionable. Below is a paper using the word in regards to this issue (though tangential to the papers main subject). Would like to hear what other think before I make the edit though. +|||||||||||||||||||||||||+ (talk) 06:18, 23 May 2012 (UTC)

http://deontologistics.files.wordpress.com/2011/03/deleuze-mmu.pdf


I think we would need more than a blog article to support saying that Zeno claimed time & space were caused by events in the brain. I think most commentators interpret Zeno to be quite sure he'd developed arguments to support "all is one" (thus, there is no plurality - there are no brainS, nor mindS... nor are brain and mind separate)--JimWae (talk) 06:53, 23 May 2012 (UTC)
I see that the epiphenomenal and Epiphenomenalism articles talk a lot about the mind and brain, following on from Descartes, and even in a medical sense, but actually I meant epiphenomenal in a more basic sense as a secondary phenomena (the perception of time) produced by a primary phenomena (matter, including everything i.e "all is one"). Therefore in this case, the secondary phenomena or epiphenomena is questionable because its secondary, an artifact. But ok, I guess its a problematic description. +|||||||||||||||||||||||||+ (talk) 08:30, 23 May 2012 (UTC)

The dichotomy paradox

  • t = 0: I am shifting the line [0, 1] to position [-0.5, 0.5]
  • t = 0.5: I am shifting the line from position [-0.5, 0.5] to position [-0.75, 0.25]
  • t = 0.75: I am shifting the line from position [-0.75, 0.25] to position [-0.875, 0.125]
  • ...

This is a list of infinitely many steps. What is the position of the line after execution of all the steps on this list, at t = 1? Is there a step on this list shifting the line to position [-1, 0]? --Netzweltler (talk) 04:21, 10 July 2012 (UTC)

Zeus vs Achilles

If you replace Achilles by Zeus, the paradox disappears, as Zeus (and only Zeus) is able to make the next step two times faster than the previous one. To calculate the elements of an infinite set, Zeus needs 10 seconds. We must recognize that the infinite sequence, the completion of which we can not imagine, however, can be completed. But only by the intervention of Zeus.

Cantor was looking at sets with Zeus eyes. Zeus had never doubt in the existence of an actual infinity.

Gilbert as the greatest of men denied the existence of the gods and, with them, the actual infinity. As a radical solution to the paradox of "Achilles and the tortoise", in the classic book "Foundations of Mathematics" (1934), Hilbert and Bernays propose to consider the space as a discrete set. For unclear reasons, the authors, unfortunately, ignore the other paradoxes of Zeno. The paradox of "Arrow" shows that the idea of space-time as a discrete set is also contradictory.

Together, the paradoxes of Zeno were aimed to justify the idea of Parmenides that our beautiful world is an illusion. Indeed, if the conflicting notions of space and time are contradictory, then there is no space-time. Then what is? There is the majestic "One" of Parmenides, Zeno's teacher, - a timeless and spaceless.

Taulalai (talk) 19:34, 23 September 2012 (UTC) 23.09.2012

Classic troll

Zeno's "Paradox of the Moving Rows" is incorrect as written on Wikipedia (or so poorly written as to appear incorrect). I found a website which presents the paradox intelligible and have included the link here, --- http://www.iep.utm.edu/zeno-par/#SSH3aiv ---. There are other websites that also present the paradox so that it can be understood, and I would be happy to include those links here if the future editor wishes. — Preceding unsigned comment added by 75.110.28.5 (talk) 01:23, 12 October 2012 (UTC)

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)

Any line segment can be divided continuously ad infinitum. At all times, with each division, the length of each part shortens by a half, while at the same time, the number of parts multiplies by two. So, how can an infinite number of parts make up a finite whole? Because for every division, the lengths of the parts are reducing as fast as the number of parts are increasing. Oilstone (talk) 21:28, 9 August 2012 (UTC)

There is not enough in sources for Zeno's writings to fully justify saying that Zeno ever actually claimed that the sum of an infinite number of numbers is infinite - though he has been interpreted that way by some. What is clear is that he thinks a task with an infinite number of steps cannot be completed.-- JimWae (talk) 01:03, 10 August 2012 (UTC)

over 10000 visits on 28 dec

Could one make a note of this here (the xkcd connection). Tkuvho (talk) 15:51, 31 December 2012 (UTC)

Misconception?

Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite.[33] However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?[4][5][6][34]

However, this "misconception" is mathematically the same as what Zeno says. "it is impossible to traverse an infinite number of things in a finite time" is equivalent to saying if we sum an infinite number of finite times, we end up with infinite time. And since Achilles is traveling at fixed speed, there is a fixed coorespondance between distance and time, and so we arrive at infinite distance. The only difference is that Zeno did not know what an infinite series was to state his objection in terms of a sum. 96.251.85.48 (talk) 05:30, 27 August 2013 (UTC)

Dichotomy paradox

"Suppose Homer wants to catch a stationary bus." As in, Homer Simpson? Can we do no better? --Thrissel (talk) 20:24, 5 July 2013 (UTC)

No. As in Homer. You know? The famous Greek epic author? 96.251.85.48 (talk) 05:34, 27 August 2013 (UTC)

Infinite Series

I've just deleted two sentences [2]. The first was an opinion - it is not task of an entry to tell what is evident, and the second nonsense. Infinite series converge. Full stop. At least some of them (The harmonic series doesn't). They do not converge with "precision", and quanta have certainly nothing to do with any mathematical proof that series converge (or not). Ansgarf (talk) 17:46, 23 September 2013 (UTC) Ansgarf (talk) 17:48, 23 September 2013 (UTC)

The last bit "particularly since there is in fact minimum expanses of time and space (quanta)" makes me think that Steaphen is back. It was added early September [3] Ansgarf (talk) 17:48, 23 September 2013 (UTC)

Merger proposal

I propose that Millet paradox be merged into Zeno's paradoxes. I think that the content in the Millet paradox article can easily be explained in the context of Zeno's paradoxes, and the Zeno's paradoxes article is of a reasonable size that the merging of Millet paradox will not cause any problems as far as article size or undue weight is concerned. Tco03displays (talk) 01:04, 4 December 2013 (UTC)

 Done
Support I see no objections. Be WP:BOLD. ;) Paradoctor (talk) 01:35, 4 December 2013 (UTC)×
It is already included in Zeno's paradoxes#Three other paradoxes as given by Aristotle--JimWae (talk) 21:04, 4 December 2013 (UTC)

Arrow paradox solution.

The amount of movement in an instant is infinitesimal, which is not zero, but it is infinitely close to zero. — Preceding unsigned comment added by Bubby33 (talk • contribs) 14:33, 21 December 2014 (UTC)

"Moving rows" paradox needs a better explanation

A reader noted that one of the paradoxes discussed the moving rows paradox, includes a diagram in a translated excerpt from Aristotle, but this excerpt falls short of articulating the paradox in an understandable way. Perhaps someone who understands the nature the paradox could add a better description, or better yet find a contemporary reliable source with a clear explanation which could be quoted.--S Philbrick(Talk) 13:48, 6 July 2015 (UTC)

Good Article

I honestly think this could be re-listed as a good article now. 75.167.203.85 (talk) 01:06, 25 July 2015 (UTC)

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Moving subsection that lacks direct support from provided source

In a subsection entitled "Hans Reichenbach" of the "Proposed solutions" section, the following appeared:

  • Hans Reichenbach has proposed that the paradox may arise from considering space and time as separate entities. In a theory like general relativity, which presumes a single space-time continuum, the paradox may be blocked.[1]
  1. ^ Reichenbach, Hans (2012) [1958]. The Philosophy of Space and Time, pp. TDB, Chicago, IL, USA: Courier-Dover, see [1], accessed 14 October 2015.

Without page number and further explanation, this source cannot be used to support this text statement. (Reichenbach discusses the circle and clock paradoxes, but does not explicitly mention Zeno.) This text therefore is either incorrect or is [original research?] as it stands. Source and explain and return this to the article, or not, as suitable. Le Prof 50.232.187.66 (talk) 13:46, 14 October 2015 (UTC)

Edits of this day

What began as a simple stop to clarify the relationship between dichotomy and the Achilles and the tortoise presentations turned into a nightmare (see preceding Talk section), as I found:

  • clear lead issues, see below;
  • structural issues, e.g., addition of sections to end without thought of overall structure, defining introductory and historical material appearing only in lead, and unsourced, etc., see more below;
  • primary vs. secondary sourcing issues rampant, i.e., repeated instances where ideas and interpretations are offered with only a primary source reference appearing, see more below;
  • consequently clear WP:OR issues, more below;
  • and, this and general scholarly sloppiness that, taken together, suggesting the possibility of plagiarism.

In short, the article as a whole repeatedly makes statements from primary sources without scholarly attribution (sourcing), and so violates WP:OR — not "possibly" OR, definite OR. Likewise, its lead makes unique statements (e.g., about Parmenides) that appear neither in later text, nor are they supported by any sources. As such, the lead is OR and is otherwise independent of, rather than reflecting, article content, as are many other points appearing later in the text (some marked). When sources do appear later in the article, they are often lacking specific enough information to be able to check contentions in the text (i.e., are lacking journal or book page numbers).

Hence, I called on a expert in philosophy to attend to the serious sourcing, original research, and possible lead and main text plagiarism issues noted. Le Prof 50.232.187.66 (talk) 14:47, 14 October 2015 (UTC)

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Wow, the lead of this article is one, overtagged mess! Anyone want to take this on? Liz Read! Talk! 14:01, 16 October 2015 (UTC)

Wasn't Archimedes rigorous?

I am not sure whether the following sentence could be scholarly (commonly) accepted: Modern calculus achieves the same result, using more rigorous methods. No doubt present-day methods are more general, perhaps even simpler, and no doubt that, say, Newton's methods cannot be considered to be fully rigorous. But I guess that there is agreement that Archimedes methods were sufficiently rigorous. In the simple case at hand, exhaustion and limit achieve the same result and it seems that they are two rigorous (different) methods. I suggest Modern calculus achieves the same result, using more general methods, but perhaps someone might dispute this (I am not an historian, either), so I am not changing the sentence, right now.78.15.206.157 (talk) 16:42, 23 March 2016 (UTC)

Arrow paradox

The arrow paradox shows the relativity of motion. If you stand on earth you see that the arrow is moving, but the arrow is motionless in a coordination-system in the centre of the system. Jestmoon(talk) 21:52, 29 May 2016 (UTC) — Preceding unsigned comment added by Jestmoon (talk • contribs) 21:48, 29 May 2016 (UTC)

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Chinese equivalents

Various IPs (probably the same person?) have been attempting to rewrite the lead to say that "Zeno's paradoxes ... were separately and independently invented and developed by both the ancient Han Chinese philosophers from the Mohist School of Logic during the Warring States period of China (479-221 BCE) and also the ancient Greek philosopher Zeno of Elea (ca. 490–430 BCE)." This has been reverted by three other editors, as at best inappropriate for the lead. Wcherowi has again revered the latest IP edit, and instead adding the following new section "An ancient Chinese philosophic equivalent":

Ancient Han Chinese philosophers from the Mohist School of Names during the Warring States period of China (479-221 BCE) independently developed equivalents to some of Zeno's paradoxes. The scientist and historian Sir Joseph Needham, in his well regarded academic work Science and Civilisation in China, describes an ancient Chinese paradox from the surviving Mohist School of Names book of logic which states, in the archaic ancient Chinese script, "a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted." Several other paradoxes from this philosophical school (more precisely, movement) are known, but their modern interpretation is more speculative.

This is better than what the IP wrote and is a reasonable first attempt at an addition noting possible Chinese equivalents (we will see if this satisfies the IPs), but it is still problematic. Here is what Needham has to say in the relevant portions of the cited source:

[p. 292:] J. P. Reding 91985), pp. 274-385, takes up the crucial question whether Hui Shih's dicta are properly understood as scientific paradoxes in the first place. He provides a [p. 293:] detailed alternative interpretation intending to show that they are not scientific paradoxes at all. A. C. Graham (1970), p. 140, on the other hand, writes: '...although Hui Shih's explanations no longer survive, the whole list can be read, like Zeno's paradoxes, as a series of proofs that it is impossible to divide space and time without contradiction.'
Unfortunately, all we have are Hui Shih's theorems and paradoxes as preserved in the last chapter of the book of his friend Chuang Tzu, to which we will now turn. The cases of Zeno and of Hui Shih differ profoundly in that we do have a fairly precise idea of the stringent logical arguments Zeno used to support his theses, whereas we know little of the intellectual context of Hui Shih's paradoxes that one can evidently raise doubt that they are scientific paradoxes in the first place, as Reding does.

These are the only references to Zeno that I can find in the cited work, and I find no reference to the "a one-foot stick" paradox, and what is there does not support the claims being made. Another relevant source (which does supply the "a one-foot stick ..." quote is the Stanford Encyclopedia of Philosophy article "Miscellaneous Paradoxes" which says:

One of the paradoxes listed in “Under Heaven” is self-explanatory, a version of Zeno's racetrack paradox:
A one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted.
Several others may be explicable in light of passages in the Mohist Dialectics, though any interpretation remains tentative.

This needs addressing. Paul August ☎ 13:39, 4 July 2016 (UTC)

Another IP has undone Wcherowi edits, and I have reverted again, pointing to this talk page. Paul August ☎ 18:29, 4 July 2016 (UTC)
Thanks Paul. I've had to revert yet again. My intention was to put the statement in an appropriate place in the article and I assumed that I was working with a good faith edit. I did tone-down some of the hype, but given your checking of the reference I now feel that even what I wrote is giving too much credence to this assertion. I don't think that the IP(s?) who are pushing this have any interest in improving Wikipedia and are only concerned with their own viewpoint. We might have to request some type of protection for the page. --Bill Cherowitzo (talk) 05:09, 5 July 2016 (UTC)

Hi people. User:Charles Matthews pinged me for my opinion on this. I think we have consensus that the ancient Chinese literature do refer to the same philosophical problem as Zeno's paradoxes. I think the real problem we have here is that we've dedicated too much of the lead to history and none to the description of the actual paradox. (I'm assuming we want the subject of this article to be the class of paradox as a concept, not a particular formation of it.) We should demote most of that history to a section called "History" and instead describe more about the paradox itself in the lead. Deryck C. 20:17, 5 July 2016 (UTC)

As conceived, and currently written, this article is about the paradoxes which have been attributed to Zeno of Elea, as referred to by many ancient authors and some of which were preserved in Aristotle's Physics and Simplicius's commentary on Aristotle's Physics. That seems the appropriate content for an article titled "Zeno's paradoxes". Not sure how that fits with wanting the "subject of this article to be the class of paradox as a concept". Note also that the title is plural, there were many paradoxes attributed to Zeno (Proclus mentions "not less than forty arguments revealing contradictions", of which nine survive, and while they share a certain commonality they are not all equivalent. As to whether any of these are equivalent to Chinese paradoxes, the only source which we have so far (mentioned above) says only "One of the paradoxes listed in “Under Heaven” is self-explanatory, a version of Zeno's racetrack paradox." which we ought to note somewhere in the article. Paul August ☎ 16:46, 6 July 2016 (UTC)

It seems clear that the IP-editor (may as well be only one) is just trying to hijack the lead instead of finding sources enough to have an article on the Chinese version where the Zeno version would have only a link or a section in the body. To have an article titled "Zeno's paradoxes" that first says "they" were developed by Chinese philosophers and then, by the way, also by a guy called "Zeno" for which there may be some piffling evidence but no "monumental academic work" is ridiculous and also kinda funny regardless of the merits. Given that this editor has no stable talk page and hasn't communicated except in one edit summary pronouncement, I also (see above and below) think the page should be protected. Debouch (talk) 16:20, 23 July 2016 (UTC)

Here is a reputable academic source that supports Chinese equivalents of "Zeno's Paradoxes"
http://www.tandfonline.com/doi/full/10.1080/09552367.2014.986934

128.90.118.145 (talk) 01:02, 3 August 2016 (UTC)

There are some problems with this source. First of all, it is a primary source for this content ... apparently some philosophical research by the author. Secondly, we can't access the actual paper (or its references) without paying. So I don't think that this falls into the category of reliable secondary sources. --Bill Cherowitzo (talk) 02:10, 3 August 2016 (UTC)

Semi-protect needed

Hi all,

There appears to be a massive amount of IP edits which frequently make edits to the article which are later reverted. This has happened many times to the article. For this reason I suggest that the article be semi-protected to prevent edits which don't contribute to the article.

FockeWulf FW 190 (talk) 15:28, 23 July 2016 (UTC)

I've just made the request to semi-protect this page. --Bill Cherowitzo (talk) 02:40, 3 August 2016 (UTC)
Thanks Bill. Materialscientist (talk · contribs · blocks · protections · deletions · page moves · rights · RfA) has semi-protected the page for one month here. Paul August ☎ 10:47, 3 August 2016 (UTC)

Nonsense paradox

The dichotomy paradox does not take time into account in that it takes M minutes to cover a set distance. It takes M/2 minutes to cover half that distance, and M/4 minutes to cover a quarter of that distance, etc. If you divide the distance up into infinitely small amounts, that just means it will take the same number of infinitely small amounts of time to cover those distances so it still takes the same time to cover the same whole distance.(5.8.184.234 (talk) 14:56, 19 September 2016 (UTC))

The moving Rows

Looks like something was mistakenly edited out here, as the paragraph no longer seems to make sense, and may be missing punctuation/capitalisation, etc. SquashEngineer (talk) 17:26, 28 September 2016 (UTC)

@SquashEngineer: The Aristotle quote is correct, see the linked cite.Paul August ☎ 17:50, 28 September 2016 (UTC)

Non-Standard (hyperreal) solutions

The turtle-like and arrow-like paradox is easily solved in Non Standard Analysis. Despite using "non-real", that is hyperreal numbers the solution holds for the "real" world.

The argument follows the general scheme that from the movement from one point to another, which are , however, at a distance of zero (0) apart from each other an infinite number of steps would be necessary, which could not be done. Let d be a real distance between start and finish. There must be infinite many points in between. An infinite number is not real, but can be written as a hyperreal number represented by the sequence [1;2;3;4;5;...] whereas a real number (e.g. 2) would be written as [2;2;2;2;...]. The infinite sequence of the infinite number surely is larger at almost all places than any sequence for a real number, hence it is realy infinite. By dividing the real distance [d;d;d;d;d;...] by [1;2;3;4;5;...] one yield the distance of two neighbouring points [d;d/2;d/3;d/4;d/5;...] which is clearly smaller than any real positive number, but also clearly larger than real zero.

Hence, in an instant - an infinitesimal small amount of time which can be represented by any infinitesimal small number like [1;1/2;1/3;1/4;...] the arrow indeed proceeds no space as [d;d/2;d/3;d/4...] is the nothingness of the distance of two neighbouring points. However, doing this flight infinite times one yields both, a real time needed to go from one starting point to a distant finish, and a real travelled distance.

The paradox is just a misconception of infinte (big) and infinitesimal (small, but not exactly zero). While the classical convergence criteria as the delta epsilon proof deliver a sound capture of defining a limit, they still do this with an aproximation to "infinite" although this quantity is nort part of the real number system. Those classical proofs still lack an universal proper treatment of infinity as they correctly state that "the usual algebraic rules" do not hold for infinity as "infinty + 1 is not more as infinity". Despite this limitation to standard rules these proofs use the operator "do this infinitively often". Weierstrass however argued correctly, as infinity + 1 shall equal infinity itself, a limit will exist if after doing something infinitively often and then do it once more the result is still the same. Non Standard Analysis prooves this to be not quite correct in the realm of hyperreals, however, the tranfer back to real numbers will exactly deliver the same result. — Preceding unsigned comment added by 114.83.130.233 (talk) 15:27, 17 November 2016 (UTC)

"Chinese equivalents" IP-editor back

An IP-editor, using different IPs each time, has edited warred, without discussion, to make this edit. See previous discussions (now archived): Talk:Zeno's paradoxes/Archive 8#Chinese equivalents "Chinese equivalents" "Semi-protect needed". I've reverted but I expect the IP-editor to continue to insist on this edit with no discussion. The page will probably need to be semi-protected again (:@Materialscientist:). Paul August ☎ 11:24, 27 November 2016 (UTC)

(@Materialscientist:) And now again. Paul August ☎ 11:58, 2 December 2016 (UTC)
And again. I concur with Paul. --Bill Cherowitzo (talk) 17:08, 2 December 2016 (UTC)

New solution proposed for paradoxes of motion

I just want to inform the community about a new solution for paradoxes of motion, which was recently published in the journal Foundations of Science :
https://link.springer.com/article/10.1007%2Fs10699-017-9544-9 (or full access here: https://philpapers.org/rec/BATWZP )
I leave contributors decide on the relevance of adding this solution in the section 'Proposed solutions' of the article.

MaelBathfield (talk) 10:35, 23 November 2017 (UTC)


Fails to mention Differential Equations and multi-variable calculus

In most forms of physics that use differential equations, an object has several properties: Mass, location, direction, and spin. Each of these properties are independent. At any incident of time, the object has a vector and a spin. Keeping track of all of the above is the function of differential equations, which show that discrete slices of an object can be frozen in time while preserving the information for the vector.

75.168.152.43 (talk) 05:48, 2 December 2017 (UTC)

Most folks just say Mass, Location, Direction, and Velocity. That's all we need to say. And anyone can look at this problem empirically and determine that the amount of time required to cover a distance, as long as it's not 0 or Infinity, becomes irrelevant. Pick any valid velocity - the problem implies there is movement - and use that number. As long as there is movement, the paradox holds and the actual velocity can be any value. There is no need to introduce differential equations here to re-describe a property that is obviously apparent in the paradox itself... There is no mathematical way to get to a remainder of zero without changing the terms of the paradox, which then proves the paradox. 73.6.96.168 (talk) 07:29, 24 November 2019 (UTC)

Lots of Math - No Proofs

I see a lot of general discussion in this article and talk page. What I haven't seen is one single mathematical proof (by a qualified person) that shows that this paradox has a mathematical solution. So, if someone would like to provide a real proof, I and others would surely like to see it. If not, why carry on with all the "mathematical" discussions about it? If there is a proof, where is it? 73.6.96.168 (talk) 07:21, 24 November 2019 (UTC)

The section "Zeno's paradoxes § Achilles and the tortoise" has a "See also" link to "Infinity § Zeno: Achilles and the tortoise" which states Cauchy's theorem addressing the matter. A footnote leads to Cauchy's proof—it isn't Wikipedia's job to provide proofs, though links to proofs should be available. Admittedly, matters should be better tied together so that concerns like yours don't arise. Peter Brown (talk) 18:58, 24 November 2019 (UTC)

Buddhist Doctrine of Momentariness

I'd like to link to a wikipedia article on the subject, but one doesn't exist. Here's a summary:

The object of the Buddhist doctrine of momentariness is not the nature of time, but existence within time. Rather than atomizing time into moments, it atomizes phenomena temporally by dissecting them into a succession of discrete momentary entities. Its fundamental proposition is that everything passes out of existence as soon as it has originated and in this sense is momentary. As an entity vanishes, it gives rise to a new entity of almost the same nature which originates immediately afterwards. Thus, there is an uninterrupted flow of causally connected momentary entities of nearly the same nature, the so-called continuum (santāna). These entities succeed each other so fast that the process cannot be discerned by ordinary perception. Because earlier and later entities within one continuum are almost exactly alike, we come to conceive of something as a temporally extended entity even though the fact that it is in truth nothing but a series of causally connected momentary entities. According to this doctrine, the world (including the sentient beings inhabiting it) is at every moment distinct from the world in the previous or next moment. It is, however, linked to the past and future by the law of causality in so far as a phenomenon usually engenders a phenomenon of its kind when it perishes, so that the world originating in the next moment reflects the world in the preceding moment.

Aero13792468 (talk) 07:57, 23 January 2020 (UTC)

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