Talk:Zeno's paradoxes/Archive 7

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Rucker

“ Rudy Rucker has mentioned a possible solution to Zeno's arrow paradox. Rucker notes that, in contrast to Zeno's assumption, the length of an arrow at rest differs from an arrow in motion due to the relativistic effect of length contraction, which Rucker says makes the state of motion "instantaneously observable". ”

Hasn't anyone challenged Rooker? Where in the paradox does Zeno say (or depend on) there is no difference at all between a moving arrow and a stationary one? It does not follow from finding a difference in length between the 2 states that the arrow can move "from where it is" to "where it is not", nor that a moving arrow (moving at constant terminal velocity in free-fall say) takes up any more or less space in one instant than in another. Does Rucker say this in a work of science fiction or in something else?--JimWae (talk) 00:07, 1 March 2010 (UTC)

The arrow, irrespective of size or speed must still move through all increments in distance. What models explain movement through and below short distances, including the Planck length, 1/2 a Planck length, 1/4 Planck length and so on?

If that model does not account for the movement of physical things, what is its relevance to this topic? What Reliable Sources confirm that relevance in detailing movement through the Planck length and shorter distances?

In other words, the entire reference/paragraph to Rucker should be deleted as being irrelevant to this matter of movement of physical things. Or at least removed and put in the section 'proposed solutions' along with Astrology and whatever else anyone feels is fairSteaphen (talk) 00:32, 1 March 2010 (UTC)

Zeno's paradoxes are not about astrology, that's why astrology is not included in the article. They ARE about a model of motion, however--JimWae (talk) 00:43, 1 March 2010 (UTC)

Astrology is a model, and by your statement models need not be rigorously matched to reality, and therefore should be considered, on that basis. Steaphen (talk) 01:21, 1 March 2010 (UTC)

If a reliable source makes the connection between the model and Zeno's paradoxes, yes. Do you have a reliable source matching astrology with Zeno's paradoxes? Gabbe (talk) 09:33, 1 March 2010 (UTC)

Do you have a Reliable Source who matches your model to the minutia of movement of physical things -things like arrows, tortoises- through and below the Planck lenghth? No? Same as Astrology, then. Steaphen (talk) 23:08, 1 March 2010 (UTC)

Rucker did not actually propose this as a solution, but just as an argument that he had never seen published. It's relevance is clear. If the special theory of relativity is reality, then motion can be instantaneously observed. Please keep in mind that this is just a claim that has been referenced. So if editors agree that it improves the article, then it definitely ought to stay in the article.

 —  Paine (Ellsworth's Climax)  00:40, 1 March 2010 (UTC)

(edit conflict) Jim, Zeno's paradox on the arrow would not say anything about a difference between a moving arrow and a stationary arrow for obvious reasons: Zeno didn't know of any difference, length or otherwise. And finding a difference in the rest length of the arrow and the moving length actually does show that the motion of the arrow is instantaneously observable, just like Rucker noted. Rucker wrote this in his Infinity and the Mind, a non-fictional analysis of "infinity", as I referenced.

 —  Paine (Ellsworth's Climax)  00:35, 1 March 2010 (UTC)

Well, 1> it is not "instantaneously noticeable", it requires measuring instruments beyond those currently available & is not directly observable at all. 2>How does Zeno's paradox depend on there NOT being a diff in length?. 3> Hasn't anyone challenged Rucker?--JimWae (talk) 00:43, 1 March 2010 (UTC)

I'm not so sure about #1, because I read that measurements of the lengths of aircraft in flight have shown this aspect of relativity to be true, and if scientists can measure a difference between the rest length and motion length of jets, then maybe they can also measure a common arrow's difference. If not, then it can always be calculated. #2, Zeno's paradox relies upon the arrow being stopped at a point in time. If motion cannot be sensed in that moment in time, then it cannot be sensed in any moment/point/instant in time. However if motion can be observed/sensed, then Zeno's arrow paradox is invalid. As for #3, I've never read of a challenge to Rucker's argument. He made it sort of "in passing" in his book, and I don't know how many people actually "caught" it.

 —  Paine (Ellsworth's Climax)  01:06, 1 March 2010 (UTC)

If I may paraphrase a few of the arguments. (1) According to relativity a moving object will be subject to length contraction. This has been experimentally observed for large object at high speeds. (2) This undermines the assumption in the arrow paradox that a moving arrow at an instant in time is the same as an arrow at rest. (3) We should probably use a different phrase than "instantaneously observed", since it might not yet be technically possible to observe it for arrows (even though it has been done for planes), and certainly not instantaneously. What Rucker want to convey that according to relativity a moving arrow has different properties from an arrow at rest. It is an interesting fact that could be included, but not core to the paradox. Ansgarf (talk) 01:17, 1 March 2010 (UTC)

"Instantaneously observed" is a quote from Rucker's book. I can put the written text in quotes, if you like.

 —  Paine (Ellsworth's Climax)  02:04, 1 March 2010 (UTC)

If it is a quote I guess it is fine then. Ansgarf (talk) 09:37, 1 March 2010 (UTC)

Irrespective of however much arrows or whatever contract, due to relativity or otherwise, what model is being proposed here that includes movement through Planck scaled increments, and 1/2 the Planck length, and 1/4, and 1/8 Planck lengths etc.

What reliable sources confirms this model is applicable at those scales?Steaphen (talk) 01:23, 1 March 2010 (UTC)

Steaphen, the model that Rucker argues for is the special theory of relativity, which has a large number of reliable sources we can use if necessary. The "scale" used is Zeno's "point in time" argument. So there is no need to invoke small increments of time and distance, especially those below the Planck length. Zeno stated that if motion cannot be observed at one point in time, then it cannot be observed at any point in time. And therefore, motion cannot be a part of reality. The special theory of relativity, on the other hand, tells us that at any point in time, the arrow in motion will be shorter than the arrow at rest. So if the special theory of relativity is a reality, then the arrow's motion can be observed by measuring or calculating the differences in length between the arrow at rest and the arrow in motion.

 —  Paine (Ellsworth's Climax)  02:04, 1 March 2010 (UTC)

Are you asserting that when an arrow flies through the air it does not pass through Planck-scaled increments, irrespective of its speed? How does the theory of relativity in any way resolve the paradox of motion, particularly for, say, a very very slow moving tortoise (relativistically speaking), and a slightly faster Achilles?Steaphen (talk) 02:11, 1 March 2010 (UTC)

No, neither Rucker nor I am asserting such a thing! The arrow must pass through all increments of time, large and small, in order to reach the target. The theory of relativity appears to solve the arrow paradox, but we must remember that the arrow paradox is about "time". It is not a "distance" paradox like the Achilles vs. tortoise paradox. So I shall have to give that one more thought. Thank you for bringing it up, though!

 —  Paine (Ellsworth's Climax)  02:25, 1 March 2010 (UTC)

Good, so tell me, what model do you, and your reliable sources propose that includes movement of arrows through and below the Planck TIME, and Planck distance?Steaphen (talk) 02:37, 1 March 2010 (UTC)

Sorry, Steaphen, I see no reason to cover old below-Planck-length ground with you. If you haven't gotten by now that there are NO scientific models that include ANYTHING below the Planck length because it's the shortest length that has any meaning, then you never will. However the Rucker claim does not deal with the need for such models. It goes right to the core of Zeno's arrow paradox and shows that, if special relativity holds, then at any given point in time, the arrow's length will be shorter when it's in motion than it is when it's at rest. The claim deals with points in time, not with periods of time that have duration. So it's a valid claim that is well-referenced.

 —  Paine (Ellsworth's Climax)  03:32, 1 March 2010 (UTC)

? If there are no models that explain below the Planck length and time, then what exactly is being proposed here, if it not based on any scientific models?

Your lack of scientific understanding beggars belief - particularly in relation to relativity which very much involves velocity, and the last time I checked elementary physics, velocity (or more correctly, speed) = d/t (that's distance divided by time). So what are you suggesting, that we ignore both matters of time and distance covered by objects in motion. Brilliant.

As for old ground, when was it ever not present ground, given that no one has presented reliable sources detailing the models that espouse movement through Planck length and shorter increments.

As for there being no models, it appears you're unaware of many interpretations in quantum mechanics which quite explicitly involve such distances, including string theory etc. Steaphen (talk) 03:46, 1 March 2010 (UTC)

(edit conflict) I grant that it's possible that there are some quantum models that work below the Planck length, but I do not know of any. Any refs. you can supply would be appreciated. Again, it is being proposed that the special theory of relativity, a scientific refinement of Isaac Newton's works, holds that at any given point in time an object in motion's length has contracted and is shorter than that object's length when it is at rest. Therefore the special theory of relativity appears to invalidate Zeno's arrow paradox. I am only arguing this with you, Steaphen, because every single word that you write tends to show that Rucker's claim is valid and belongs right where it is in this article.

I will not comment on your feelings about my scientific understanding, as I prefer to assume good faith on your part, and any comment I make would work to invalidate that preference.

 —  Paine (Ellsworth's Climax)  04:12, 1 March 2010 (UTC)

Paine, irrespective of what Rucker says or explains, any theory which does not include movement through Planck scaled increments becomes irrelevant to the precise solutions to Zeno's Paradoxes, which only again confirms the validity and appropriateness of 'high-accuracy' and no more than 'high-accuracy'.

As for offering models, not interested, at all. Not my job. Sorry. Steaphen (talk) 04:27, 1 March 2010 (UTC)

Your assertion does not hold true when it comes to Zeno's arrow paradox, which deals solely with points in time. It does not deal with movement through "scaled increments", only points in time. As for offering models, if you don't want to back up your statement, s'okay with me. Lends little to the credibility of your stance, though.

 —  Paine (Ellsworth's Climax)  04:46, 1 March 2010 (UTC)

Paine, see below. As for my credibility, assume I have none, whatsoever. Begin there, with your assumptions. Then look to the questions that I ask, to find whatever credibility is needed for your stability and peace of mind.Steaphen (talk) 04:54, 1 March 2010 (UTC)

As before, the Dichotomy Paradox involves movement through increasingly smaller (1/2 sized) increments of distance. The arrow, relativistically speaking or otherwise, involves movement through Planck-scaled distances. The onus is upon those making assertions of supplying credible Reliable Sources who address the issue of movement of Zeno's Homer, runner, arrow and tortoise through ALL increments, including those at and below the Planck length, half a Planck length, a quarter of a Planck length, one hundredth of a Planck length, one millionth of a Planck length etc. This applies to all the paradoxes. Measurement is irrelevant. This is a conceptual issue.

Upon what basis does one justifiably reject consideration of movement through and below the Planck length?

What reliable sources confirm the validity of that rejection, and why?

Until confirmed otherwise, the 'high-accuracy' remains both valid, appropriate, and the most scientifically credible.Steaphen (talk) 03:56, 1 March 2010 (UTC)

Let me give you a gentle reminder that this section is about Rucker's statements, and he made no statements about any other paradox of Zeno's, only about the arrow paradox. And while the first two paradoxes in the article start by dividing space, the arrow paradox starts by dividing time, and it divides time not into intervals, periods or durations, but into points of time. I have no qualms about the "high-accuracy" statement. My quarrel is with your continued entrance of short intervals of time into the arrow paradox, which is solely about points in time. Please reread the arrow paradox section of the article. You do not seem to have a good grasp of it yet.

 —  Paine (Ellsworth's Climax)  04:22, 1 March 2010 (UTC)

is a point in time of infinitely short duration? in which case getting to that point - via you getting there, or the photons that allow you to observe the arrow at THAT point, must travel through those Planck-scaled increments, correct?

Good, glad to see we're getting there! (ah, another pun, that I don't mind saying is not so bad). Steaphen (talk) 04:48, 1 March 2010 (UTC)

(edit conflict) Glad to see that you're glad, Steaphen! A point in time is of zero (0) duration, and it is my understanding that there are some philosophers who would call that an "infinitely short duration". This is not about the actual motion of the arrow, though. Zeno stopped the arrow in time, and said that at that point in time, the arrow cannot be in motion. It is not moving into that position, and it is not moving out of that position. Therefore at that point in time, the arrow must be motionless. No argument. THEN, Zeno goes on to say that if the arrow is motionless at that one point in time, then it has to be motionless at every point in time. Ergo, the arrow cannot possibly move. And until the special theory of relativity, there was no way to prove Zeno wrong about the motionless arrow. IF the special theory of relativity holds, then it shows that at any given point in time, an arrow in motion is contracted and shorter in length than that same arrow when it is still in the quiver. This is a good test for the theory, and the test has been performed, and the special theory of relativity passed this test. Therefore, in all liklihood, Rudy Rucker is correct, and the special theory of relativity soundly invalidates Zeno's arrow paradox.

 —  Paine (Ellsworth's Climax)  05:05, 1 March 2010 (UTC)

Dear Paine, by all means carry on with your points in time. But the moment (no pun intended) you mention 'motion' you involve increments in time and space (motion, speed thereof = distance/time). Thus, Rucker does not soundly do anything of the sort. UNLESS the arrow is flying at the speed of light, and has therefore shrunk to a point, with infinite mass. But let's not even begin to go there (awh, I can't help meself with those puns)Steaphen (talk) 05:21, 1 March 2010 (UTC)

There are good sources that explain Rucker's view, namely his own book. Since it is sourced we can include a description of Rucker's argument. If there is a source that disagrees with Rucker, we should include it as well. If we conclude that Rucker's argument is invalid, but have no source to back it up, it is unfortunately an original contribution. Ansgarf (talk) 05:25, 1 March 2010 (UTC)

Tell me, good gentle people, when do the 'Zenoan' Paradoxes NOT involve increments in either time or space? If you wish to assert that 'points in time' do not involve increments, how might we ever expect to verify that assumption, that we do not move through increments in time? Does not the mere fact of thinking involve increments in time? What pray tell doesn't involve increments in time and/or space in regards to these paradoxes?

Assume that I'm a dummkopf, and need speaking to as a seven five year old, who doesn't 'get it'. :) Steaphen (talk) 05:03, 1 March 2010 (UTC)

I thought I'd already done that! Well, so much for my "scientific understanding", as well as my instructional ability, eh? Steaphen? If you really and truly need such a basic explanation, then why aren't you reading the most basic of references? In effect, why are you here? IOW, why are you bothering to argue such esoteric subjects as Zeno's paradoxes if you can't even grasp the simplest of explanations, which I've already given you?

 —  Paine (Ellsworth's Climax)  05:23, 1 March 2010 (UTC)

you 'thought'? at what point in time did you do that?

As for why am I here? Better still, how do you think I got here? But before attempting to answer that, remember those Plancks in your eyes.

Are we having fun yet? Ok then, let the fun begin, 'explain how anything moves through consecutive points in TIME without also moving through Planck-scaled increments in space.' (recognising here that, at least on this planet -not sure what planet you lot are on- everything is continually hurtling through space at a fair clip).

btw, this is better than any entertainment you'd pay for.Steaphen (talk) 06:06, 1 March 2010 (UTC)

While there is no disagreement from me that a moving arrow has a slightly diff length than a stationary one, I still must disagree that Rucker's off-hand comment in any way defeats Zeno's argument. According to Zeno, all instantaneous velocities are zero since at any instant the position of all objects is not changing. At that instant the arrow would occupy a certain space (it is where it is and not anywhere else). If its instantaneous velocity is zero, then it cannot be moving at all and cannot move to another place. The arrow paradox depends on the instantaneous velocity being zero, something we, of course, disagree with. We do not prove the instantaneous velocity is not zero by measuring the length during any instant - because neither length nor velocity can be measured without taking more than an instant of time to do so. It is only theoretically speaking that we can talk of the "instantaneous length" of anything - it cannot be observed. While Rucker's point is defintiely interesting & relevant to theories of motion, putting this apparently unexamined and apparently offhand comment in the section about the status of the paradoxes today is to assert that it is cited or discussed by others in the field. I do not disagree that we find a place for it in the article, but where to place it is problemmatic. Perhaps we need another section or a different title--JimWae (talk) 09:21, 1 March 2010 (UTC)

And, Jim, if we disagree with the instantaneous velocity being zero, then we must agree that if SR holds true, the arrow will be a shorter length, thereby proving that the arrow is in motion, thereby defeating Zeno's argument. I agree that a measurement of the arrow at any given instant would be a monumental, if not impossible, task. I appear to have been wrong about the measurement of the length contraction of jet planes. What I had actually read was something like this. At any rate, I agree that the "Status . . ." section is inappropriate for this claim. So I treated it as a "Proposed solution" by Rucker and put it in that section. Problem solved?

 —  Paine (Ellsworth's Climax)  19:52, 1 March 2010 (UTC)

I agree with JimWae. That's two things we've agreed upon. The planets must have realigned. The Rucker thing is largely a side-show, and as originally suggested, should be in another section, if any at all.Steaphen (talk) 09:39, 1 March 2010 (UTC)

(edit conflict) I'm unsure if it should be in the article at all. There are probably a gazillion off-hand comments made by various sources, why single this one out? It seems WP:UNDUE. I think the article would be much improved by sticking with mainstream arguments made by the most respectable academic sources, and frankly, ignoring sources like Rucker. Gabbe (talk) 09:46, 1 March 2010 (UTC)

As said, the Rucker quote is an interesting observation that invalidates one of Zeno assumptions with a novel argument, but is not really at the heart of the Zeno argument. It might very well be WP:UNDUE. We could move it to the popular culture section. Ansgarf (talk) 11:35, 1 March 2010 (UTC)

I agree that the proposal by Rucker was probably WP:UNDUE while it intruded upon the "Status . . ." section. However, as a proposed solution by a published author and scientist, it couldn't be constituted as WP:UNDUE in the "Proposed solutions" section, could it? If you still think it grabs too much attention, let me know, because if this is the case, it might be better to take it out of the {{Quote}} template and just tack it on to the "Rucker maintains". As for mainstream, what is more mainstream than the special theory of relativity? Rucker's comment was not "off-hand" at all. It serves as a possible solution to one of the paradoxes, so doesn't it deserve the little mention it has been given?

 —  Paine (Ellsworth's Climax)  20:17, 1 March 2010 (UTC)

Are there any reliable sources (or any other sources at all) mentioning his solution? If not, his opinion is most likely (at best) a minority viewpoint among reliable sources, and in that case mentioning it in the "Proposed solutions" section would probably count as "undue weight". Gabbe (talk) 21:04, 1 March 2010 (UTC)

A search titled "zeno's arrow paradox solution rucker" yields Rucker's mention alongside the arrow paradox in several works:

• Further reading: [1]
• Mentioned alongside Bertrand Russell: [2]
• Referenced: [3] – [4] (in Peter Suber's bibliography) – [5] (scholar search of Google)

(HTH)  —  Paine (Ellsworth's Climax)  22:33, 1 March 2010 (UTC)

Regrettably, none of the above refers to Rucker's remark on the implications of relativistic contraction for the arrow paradox. The links do show that Rucker is read in academic circles, which is good enough to justify keeping him in the article. I'll add Rucker, Sewell, Suber and Verelst to the bibliography, and citations of the Rucker book to the Rucker entry. Paradoctor (talk) 00:06, 2 March 2010 (UTC)

Changes

Practically this entire section constitutes WP:FORUM. It started with a good question, and immediately veered off into discussing the topic. I'm happy with the current state. If anybody wants changes, please state what you want changed. Paradoctor (talk) 22:14, 1 March 2010 (UTC)

• Concur.  —  Paine (Ellsworth's Climax)  22:33, 1 March 2010 (UTC)

• I think putting the Rucker quote in the text is WP:UNDUE (because 1>apparently no scholar has found it relevant enough to discuss at length, but also because 2>"instantaneously observable" is so problematic). I have no problem with putting it in the ref--JimWae (talk) 01:41, 2 March 2010 (UTC)

Done. Paradoctor (talk) 02:08, 2 March 2010 (UTC)

The fact that a source has not yet been found specifically pointing to conversations about Rucker's assertion does not mean necessarily that such conversations have not taken place. It is a valid claim that is well-sourced and should be in the article. And I will continue to try to find a focused third-party reference.

 —  Paine (Ellsworth's Climax)  02:03, 2 March 2010 (UTC)

"not mean necessarily": Agreed, but sooner or later, we need to prove our assertions. If nobody else objects, the current state of the Rucker mention seems to have consensus.

"try to find a focused third-party reference": Very good, that's the kind of contribution that makes Wikipedia grow like weed. ;) Paradoctor (talk) 02:15, 2 March 2010 (UTC)

Finally noticed your edit. Please explain your edit summary, I don't see how MOS:QUOTE applies. Also, you might want to address JimWae's objection. Paradoctor (talk) 02:52, 2 March 2010 (UTC)

Self-reverted. And MOS:QUOTE guides us to use the Quote template under certain conditions. Not all conditions were being met.

 —  Paine (Ellsworth's Climax)  10:58, 2 March 2010 (UTC)

This is more or less moot now, but I think you misread MOS:QUOTE here. It only says you should use <blockquote> when the quote is big. That does not mean you can't use it for smaller quotes. Oh well, if everybody is content with the paragraph as it is now, let's count our blessings. ;) Paradoctor (talk) 11:30, 2 March 2010 (UTC)

I'm counting, definitely counting. I wasn't being clear enough on the MOS:QUOTE issue, as I was also thinking about how putting a short quote like Rucker's in the Quote template might seem to some readers to imply that Wikipedia endorses the claim. This is, I suppose, a form of violation of WP:UNDUE. As you know, Wikipedia does not endorse any claims made in any articles. Wikipedia remains a source of information only. Thank you, Paradoc, for aiding this inexperienced editor! (me)

 —  Paine (Ellsworth's Climax)  09:41, 3 March 2010 (UTC)

Please don't get me wrong, but are you being sarcastic? Your edit count is almost thrice mine! Puzzled, Paradoctor (talk) 23:09, 3 March 2010 (UTC)

No, no! I have found that Wikipedia editorial experience is a whole lot more than just total edit count. I have been editing WP for many years, mostly as an IP, so my edit count is probably thrice what it shows to be. However, I still learn new things about editing almost on a daily basis, and sometimes from editors with a lot less experience than I have. So no, absolutely not, no sarcasm intended.

 —  Paine (Ellsworth's Climax)  17:07, 4 March 2010 (UTC)

"still learn new things about editing almost on a daily basis": Hey, wasn't that my line? ^_^ Seriously, I'm afraid you'll have to finally face it: You are a WikiGrownUp now. "Inexperienced", tsk tsk... ;) Paradoctor (talk) 21:14, 4 March 2010 (UTC)

Guess I'm one hekuva plagiaristic SOB, then, huh. < grin > About my being a WikiGrownUp, just know that I often seek to satisfy my WikiInnerChild. I like to "let him loose" every now and then so that, at age 12 (lustra), I still feel young.

 —  Paine (Ellsworth's Climax)  22:18, 4 March 2010 (UTC)

While more modern calculus has solved the mathematical aspects of the paradox with high-accuracy

(strikethroughs added by Paradoctor (talk · contribs), please see my reply below)

Ansgar has again posted without credible Reliable Sources supporting his edit.

In detail, he maintains that calculus solves the mathematical aspect of the paradox with perfect accuracy, not fair, good or with 'high-accuracy' as was agreed upon by others.

What Reliable Source confirms that calculus is a valid, experimentally verified 'mathematical aspect of the paradox' through ALL scales, including at and below the Planck length?

(in the interests of consensus and legibility, content that was 'struck-through' by Paradoctor, removed by Steaphen :)Steaphen (talk) 02:46, 2 March 2010 (UTC)

You and Ansgarf have a content dispute. If you revert without first reaching WP:consensus with Ansgarf and me and JimWae and Paine and whoever else is contributing here, you WP:EDITWAR. It doesn't matter who started, during consensus-seeking there will always be a time when the article displays the WP:WRONGVERSION.

Please do your best to provide WP:DIFFs, I had to dig it up from the article history.

In case you wonder about the strikethroughs, you can remove them if want. They are only meant to indicate to you the parts of your message that I consider inappropriate and counterproductive in here. If you disagree, I'll gladly detail my reasoning to you.

As regards your adding of the modifier "with high accuracy", please provide a source stating exactly that. I presume you will agree to be held to the same standard you hold Ansgarf to? Paradoctor (talk) 00:36, 2 March 2010 (UTC)

All of the quoted sources in that statement tell that the mathematical aspects have been solved, some even show how to solve them. None of them uses the qualifier "with high-accuracy". Ansgarf (talk) 00:44, 2 March 2010 (UTC)

I agree about the modifier, but please note that providing calculations is not the same as saying "that the mathematical aspects have been solved". One may disagree, but the statement has been contested, and now it is necessary to WP:PROVEIT.

I have repeatedly asked a simple question (and let the implications of the failure to allow questions weigh on each of you): "To what degree does calculus solve the paradoxes ... with poor-accuracy, high-accuracy or perfect accuracy?" If you cannot confirm 'perfect' accuracy, then include the extent to which it can be verified by present known science.

As the record shows, a compromise was agreed, with the initial agreement to read 'fair-accuracy', which was later amended, after objection by JimWae, to read 'high-accuracy'. Ansgar subsequently reverted that which was agreed (to use 'high-accuracy'), without consensus Steaphen (talk) 01:41, 2 March 2010 (UTC)

"agreement": Did Ansgarf agree? If not, that was not a consensus edit. Please state whether you intend to supply a source supporting the modifier. I am challenging that modifier, and will remove it if no source supporting its use is provided.

"simple question": Wrong place to ask, we're an encyclopedia. The right question to ask is: "What does the literature say?" Paradoctor (talk) 01:54, 2 March 2010 (UTC)

"Ansgarf (and I) have a content dispute. If you revert without first reaching WP:consensus with me and JimWae and Paine and whoever else is contributing here, you WP:EDITWAR. It doesn't matter who started, during consensus-seeking there will always be a time when the article displays the WP:WRONGVERSION."

As for the rest, 'tis a display of poor reporting by you. You will note the reference to 'agreement', which does not imply or state full agreement or 'consensus.' and the reference to 'consensus' was in the context that Ansgar did not revert with 'consensus' (i.e. full agreement). Thus, the sentence is technically and grammatically correct. Steaphen (talk) 02:32, 2 March 2010 (UTC)

You're not really trying to pull "technically" with me? On Wikipedia, to boot? It doesn't matter who you agree with, be it JimWae, be it JimBo. As long as somebody disagrees and that somebody's concerns are not properly addressed, there is no consensus. Plain and simple. Paradoctor (talk) 03:02, 2 March 2010 (UTC)

The sentence, as posted, is correct. Agreement was reached, period. I did not state 'full agreement'. What part are you having difficulty with, technically speaking?Steaphen (talk) 03:08, 2 March 2010 (UTC)

Whoops! awfully remiss of me. Ansgar did agree. "Until others give their input, I am happy to leave the 'high-accuracy' in there. No worries.Ansgarf (talk) 07:48, 28 February 2010 (UTC)" Perhaps then I should have said 'consensus', in lieu of the remaining 6 or so billion still to comment? Steaphen (talk) 03:13, 2 March 2010 (UTC)

In my comment I meant to say that I can live with the "wrong version" until we get feedback from others. Now we have feedback from others. Ansgarf (talk) 03:41, 2 March 2010 (UTC)

(@Steaphen) To quote myself: Do "you intend to supply a source supporting the modifier"? Ansgarf and I oppose it. JimWae's acceptance of it seems to have been motivated by trying to get some workable compromise, rather than belief that it represents the literature. That's of course only my guess. Paradoctor (talk) 11:56, 2 March 2010 (UTC)

I have, to my knowledge, never disallowed or dismissed the technical mathematical correctness of anything on these talk pages, only its direct relevance to the issue of physical movement (aka the problems posed by Zeno of Elea). By all means, perform your calculations and come to whatever conclusions you like, but PROVE that calculus actually correlates with what you say it does in relation to physical movement through ALL scales of movement. If you can't prove it, the statement 'more modern calculus has solved the mathematical aspects of the paradox with high-accuracy' remains the most scientifically credible and testable! Steaphen (talk) 04:04, 2 March 2010 (UTC)

"PROVE that calculus actually correlates with what you say it does": That's a fundamental misunderstanding of what this encyclopedia is about. We do not prove anything, we report on proofs in the literature, at most. Anything that is not sourced is just a polite, and temporary, compromise.

"the statement 'more modern calculus has solved the mathematical aspects of the paradox with high-accuracy' remains the most scientifically credible and testable": Cite that to reliable sources, and I'll personally make sure it sticks to the article. Paradoctor (talk) 11:56, 2 March 2010 (UTC)

The original statement that "more modern calculus has solved the mathematical aspects of the paradox" was contested by me, and therefore requires that you WP:PROVEIT.

As a compromise I was willing to accept fair-accuracy, and at a stretch 'high-accuracy'. However, I'm now disinclined to accept any compromise. If you can't prove that statement, that 'more modern calculus has solved the mathematical aspects of the paradox" then please remove it entirely. That proof will need to explicitly confirm that "the mathematical aspects" are actually (as confirmed through evidence) "OF the paradox", which includes the detail of movement through increasing smaller increments, including the Planck length and infinite orders of magnitude shorter. Steaphen (talk) 13:02, 2 March 2010 (UTC)

I take that as a sign that you won't oppose removing the modifier. I think Paine won't mind either, which appears to mean that we have consensus. Removed it. I have tagged the remaining statement as {{page needed}}, as I am opposing immediate removal. If not satisfied, it will have the same effect as an unanswered {{cn}}, leading to its removal. Paradoctor (talk) 13:09, 2 March 2010 (UTC)

Without proper sourcing, I cannot oppose the modifier's removal. It appears that editor Steaphen is seeing WP:PROVEIT (aka WP:BURDEN) in a way that was not intended. This might be a fault of the content of that section of WP:V. Perhaps it can be improved. Anyway, Steaphen, that section of WP:V does not deal with whether or not a claim is "true". It only deals with the burden of evidence for the claim. For example, the burden of evidence for the discussed calculus claim has been met; however, the burden of evidence for the "high accuracy", or any comment about the level of accuracy/precision, has not been met.

 —  Paine (Ellsworth's Climax)  10:07, 3 March 2010 (UTC)

Numerous calculus sources state that the paradox simply is resolved using limits. Here are some I found quickly; I am sure there are more:

• "The resolution of the paradox is that although the number of time intervals being considered is infinite, the sum of their lengths is finite, Achilles can overtake the turtle in a finite time" Calculus II, Jerrold E. Marsden, Alan Weinstein, p. 568
• "Even if the solution to Zeno's paradox using limits seems unnatural at first, do not be discouraged. It took over 2,000 years to refine the ideas of Zeno and provide conclusive answers to those questions about limits that will be presented later in this chapter." Calculus, By Gerald L. Bradley, Karl J. Smith, p. 76
• "In other words, Zeno's paradox of the Achilles is to be answered in precisely such ideas, based upon the limit concept" The history of the calculus and its conceptual development, Carl Benjamin Boyer, p. 281
• "What Zeno objected to was infinite divisibility. We overcome this problem with the concept of convergence, basing it on our formal definition of limit. In this section we will consider sequences and series of constants and allow Achilles to ultimately catch the tortoise." Introduction to Real Analysis: An Educational Approach, By William C. Bauldry, p. 25

I still recommend splitting the "modern status" section into two: one to cover the mathematical solution, and one to cover the lingering philosophical concerns. — Carl (CBM · talk) 13:09, 2 March 2010 (UTC)

These quotes do not constitute EVIDENCE for the statement "modern calculus has solved the mathematical aspects of the paradox". The theory of limits is a theory. Where is the evidence, experimentally verified, and confirmed by Reliable Sources? Steaphen (talk) 13:31, 2 March 2010 (UTC)

"is resolved using limits": Then we have to say that, if the other formulation is contested and can't be sourced.

"splitting": Please, one issue at a time, I'm still catching up. Paradoctor (talk) 13:14, 2 March 2010 (UTC)

The present structure of the article makes things difficult. On the one hand, from a purely mathematical view, using Newtonian mechanics, we have no difficulty resolving the paradoxes. But at the same time there really is a substantial body of literature that says that this mathematical solution is not philosophically satisfactory. These are separate issues and not actually in conflict. — Carl (CBM · talk) 13:24, 2 March 2010 (UTC)

"present structure of the article": You're too generous. I have announced a review of the article before I went AWOL, and I mean it. Of course, as long as the current ruckus is not settled in some way, I get neither the time nor the motivation for a sizable effort like that. :(

"separate issues and not actually in conflict": I'm pretty confident there are sources disagreeing with that. Regardless, the issue at hand is a specific, contested, and as yet unsourced formulation. Paradoctor (talk) 13:44, 2 March 2010 (UTC)

When sources say "the calculus solution does not resolve the paradox" this doesn't mean that they are actually in disagreement with the calculus solution,they just think it does not address the actual paradox. Of course there are also many authors who say the calculus solution does address the paradox. These authors are simply looking at different aspects of the paradox.

In any case, I posted the sources above to give some indication that there is support in the literature for the claim that calculus does resolve the paradoxes, since for some reason the existence of such sources appeared to be in question (although anyone who has looked at a few calculus textbooks will know Zeno's paradoxes are a standard example). — Carl (CBM · talk) 14:13, 2 March 2010 (UTC)

The current article does currently already distinguishes between the philosophical aspects, and the mathematical aspects, and those then again separate from the aspects of physics. We could make this distinction more pronounced. Mathematical and philosophical aspects are either solved or not, but not solved with "high accuracy". There might be disagreement among philosophers whether they are solved though. Only aspects of physics can be solved with "high accuracy" since they involve measurement and uncertainty. It would be fair to say in the article that the mathematical aspects have been solved - for which we have sources - while there is disagreement whether this solves the philosophical aspects -for which we have sources too. Ansgarf (talk) 21:58, 2 March 2010 (UTC)

That is exactly my opinion about how the article should be organized. — Carl (CBM · talk) 22:18, 2 March 2010 (UTC)

I agree as well. Paul August ☎ 22:23, 2 March 2010 (UTC)

Inaccurate statement:

While more modern calculus has solved the mathematical aspects of the paradox with high-accuracy

it cannot be expressed so. Maths like logic, solves problems "exactly" or simply put it solves problems STOP, not solves it with "high-accuracy", which erroneously indicates that the solution is a numeric mostly-solution. Instead a correct statement would be:

Mathematical real analysis provides the answer to the infinite iteration by giving a definite limit as an answer to that iteration

which is an algorithmical answer: the iteration process is flawed, but when observing it from outside, we can see that it converges to the limit value where f.ex. Akilles meets the turtle. That answer does however not regard the reification fallacy invoked when equating the observers flawed iteration process with the world itself. IMHO the paradox is solved for all reasonable ontology systems. Rursus dixit. (^mbork³!) 08:37, 12 June 2010 (UTC)

AN/I thread

A thread related to this article has been opened at Wikipedia:Administrators' noticeboard/Incidents#User:Steaphen. Nsk92 (talk) 23:50, 2 March 2010 (UTC)

The thread moved to the Administrators'_noticeboardAnsgarf (talk) 02:23, 3 March 2010 (UTC)

Oh-kay. I hope the block affords Steaphen the opportunity to ponder the possibility that he has misunderstood a couple of important aspects of Wikipedia, and will have a better experience when he comes back. Meanwhile, I'll tilt my head a little, let the warm light wash over my face, and enjoy the sudden silence for a while before I get back to work. :) Paradoctor (talk) 23:18, 3 March 2010 (UTC)

That takes care of ArbCom. Let's hope we can ignore the existence of WP:DR for some time to come. Paradoctor (talk) 22:04, 8 March 2010 (UTC)

Status of the paradoxes today

I think I just realized why this section attracted so much trouble: It doesn't belong in an encyclopedic article! For one, it is clearly incompatible with WP:DATED. Also, in order to satisfy its goal, we'd need a comprehensive review of the current literature, and one may confidently expect that review to become out of date itself soon, if it even exists. I suggest we kill the section and distribute its content elsewhere. Paradoctor (talk) 14:13, 6 March 2010 (UTC)

Good idea. Paul August ☎ 14:39, 6 March 2010 (UTC)

I seem to be missing something. I don't see anything about this section where WP:DATED would apply. "Today" to me means "modern times", so how can we remove the whole section? Seems to be important information that improves the article to me. I say we keep it per WP:PRESERVE. After reading about the interesting subject of Zeno's paradoxes, I think readers will want to know as a next step in their knowledge just where those paradoxes have "landed" in the modern day.

 —  Paine (Ellsworth's Climax)  16:14, 6 March 2010 (UTC)

• PS. If they have landed in the deep pit of Wikieditorial controversy, that is no reason to apply TNT to the section.
• PPS. I have renamed the section to clarify. I will also add an expansion tag. One reason such things are troublesome is that not enough information is given.

"where WP:DATED would apply": "today"? I'm not against a section on current status, but then this section needs to delineate start and end of the era covered. In 100 years, reception of Zeno might be substantially different than what it is "today". And the start is far from obvious, either. Are we talking "this generation", "this century" (i. e. since Einstein), since Newtown, since Galilei, post-Renaissance? Is there a actually something like identifiable epochs in the history of the debate?

Don't drag llamas into this, or I'll whip out my squirrels. ;) Paradoctor (talk) 23:30, 6 March 2010 (UTC)

I agree that the section needs work. "Today" or "modern times" is admittedly subjective and could be defined better by the section's content. At present, the time period involved seems to be loosely defined by the reference sources. As with many controversial items, debate often has its ups and downs in terms of intensity, and I'm not really sure what epochs this debate inhabits. The section and article could definitely benefit by the entrance of an expert.

And I promise you Paradoc, no llamas. I have more than enough squirrels climbing my leg for peanuts.

 —  Paine (Ellsworth's Climax)  00:46, 7 March 2010 (UTC)

Uh, err, umm, ok! That's probably more than I ever wanted to know about your personal habits. ;) Paradoctor (talk) 15:42, 8 March 2010 (UTC)

Zeno specifications

"Leslie Lamport's Specifying Systems contains a section (9.4) introducing the character of the Zeno Specifications": Was in the "Writings" section, needs source before going back into the article. Paradoctor (talk) 14:28, 6 March 2010 (UTC)

 Done – Odd though, that I didn't notice at least a CN template, or perhaps even your effort to locate a source yourself. (It was really easy, as I found it in the Leslie Lamport article.)

 —  Paine (Ellsworth's Climax)  19:06, 6 March 2010 (UTC)

Whoops, should've stated more clearly my problem. I don't think it belongs in "Writings about". Zeno specifications is, like Zeno behavior and quantum Zeno effect, a concept related to the paradoxes, but not about them. Had a look a the reference, and it looks like it belongs in Zeno behavior.

"your effort to locate a source yourself": I love it when you say rude things to me. :-P Paradoctor (talk) 23:42, 6 March 2010 (UTC)

I must've forgotten to take three deep breaths and to count to ten before responding! <g>

 —  Paine (Ellsworth's Climax)  00:51, 7 March 2010 (UTC)

Moved Lamport ref to where I think it belongs. It is also more explicit about excluding Zeno behavior, and more general too.Ansgarf (talk) 03:08, 7 March 2010 (UTC)

Agree  —  Paine (Ellsworth's Climax)  05:40, 7 March 2010 (UTC)

Scar tissue?

Some of this stuff reads oddly to me - e.g. Zeno's paradoxes remain a problem for philosophers.[5][6][7] Variations on the paradoxes (see Thomson's lamp) continue to produce philosophical problems. but then I'm not a philosopher. Mathematicians certainly don't care. This isn't scar tissue left over from the Big Fight is it? Ditto the POV header and text in "The paradoxes in modern times" - though maybe that will disappear entirely, per above William M. Connolley (talk) 20:35, 6 March 2010 (UTC)

Hi William! I'm a little confused, first that no one else has answered you, because I'm certainly no philosopher either, and secondly by some of your concern. When it comes to Zeno's paradoxes, it appears that the two schools of thought, math and philosophy, just don't see eye-to-eye and never have. It does seem odd that such obviously incorrect (if we are to judge by our senses) conclusions would still be with us now, but they are. In spite of the ability of math to deal with the paradoxes and to show (as far as the mathematicians are concerned) proof that they are wrong, philosophers are as yet, uhm, not impressed, not convinced that the paradoxes don't go much deeper than mere math can tread.

Why do you think that "The paradoxes in modern times" is a POV header? I don't see it as any more POV than, say, "History of the paradoxes". Don't you think that readers would find such a section as a sort of culmination of the article? Admittedly, as I said above, the section does need work, and I'm hoping that some philosophy expert will come along and help the article with his or her knowledge. Until that happens, there are valid claims in the section that are reliably sourced, so they have a place in the article. Any POV content does need to be improved so that the section is neutral as possible. Anything you can do to aid in this effort is very welcome!

 —  Paine (Ellsworth's Climax)  23:11, 7 March 2010 (UTC)

'Why do you think that "The paradoxes in modern times" is a POV header? - ah, I don't, I was ambiguous: I think the section and header is OK, I can see no reason for a POV tag on it. So I took it out.

As to the philosophers: the maths is within my grasp, the philosphoers troubles isn't, so I was wondering how much the emphasis on philosophical problems was perhaps over-emphasised. Is it true that all (relevant) philosphers are troubled? Or is it perhaps just a minority view within the philosophy community? William M. Connolley (talk) 14:33, 8 March 2010 (UTC)

Philosophers are always troubled. The current of state of the article doesn't bother me overly. Before our bibliography is not at least up to the standard of Salmon's, our coverage will be spotty at best. What I am saying is, edit whenever you feel competent and motivated, but when in doubt what to do, focus on collecting and criticizing sources right now. Paradoctor (talk) 15:38, 8 March 2010 (UTC)

Sorry I misunderstood you! One area of inquiry might be the idea of infinities and how they apply here. Philosophers seem to think that mathematics doesn't really have a good handle on the concept of "infinity" in the sense that Zeno inferred by way of his paradoxes. There just isn't enough "room" between 0 and 1 to hold all of the troubles philosophers have with these ever-interesting, ever-intriguing paradoxes. So no, I don't think the philosophy side is over-emphasized. If anything, it's probably understated. The math seems pretty straightforward and well-covered. It's the philosophy side that needs expertise and expansion.

 —  Paine (Ellsworth's Climax)  06:42, 9 March 2010 (UTC)

Perhaps it is the philosophers with a problem with the mathematical solution of Zeno's Paradoxes who don't have a good handle on the concept of infinity, since both mathematics and reality both suggest that there is no problem, except maybe with those same philosophers. I think the people who argue that Zeno's Paradoxes have not been solved are the same types that will argue that $0.99{\overline {9}}\neq 1$.

I also fail to understand how you have come to believe that the mathematics are well-covered. I don't see a single solution to a single paradox in the article itself.

Prophet of nuggan (talk) 16:47, 9 March 2010 (UTC)

Well, Prophet, some people seem to feel that to actually place math solutions in the article would be in violation of WP:NPOV, since more emphasis is already needed on the philosophical content. The math is covered in the reliable sources. What is needed in that short section on the paradoxes in modern times is more detail on what philosophy is doing with these paradoxes. Are they still trying to solve them? Have they thrown up their arms and given up? Details, details, details. This is what would make the section more in line with a neutral point of view.

 —  Paine (Ellsworth's Climax)  14:30, 11 March 2010 (UTC)

"Perhaps it is the philosophers" ... "who don't have a good handle on the concept of infinity": If you can source this to reliable literature, you're more than welcome to add it to the article. Paradoctor (talk) 18:03, 9 March 2010 (UTC)

Agreed. The math is completely straightforward. But what does the math show us and what does it do as far as solving the paradox goes? Let's see. The math can show us things like where and when Achilles will pass the tortoise. But Zeno didn't care about knowing that. He wanted to know how it is possible for Achilles to pass the Tortoise in the first place: showing him that Achilles does pass the Tortoise at some time and place doesn't answer that question. Zeno knew that this was possible, he didn't know *how* this was possible. Using math to calculate where and when Achilles passes the Tortoise is therefore really not any different from having them race in the real world, thereby showing that Achilles does pass the Tortoise: doing that doesn't *solve* the paradox, rather it *creates* the paradox!!! To be precise, both the real world and mathematics tell us that Achilles will pass the Tortoise, but Zeno's argument concludes that that is impossible: *that* juxtaposition of two contradictory claims is the paradox!

Another thing math tells us that the sum of an infinite number of terms can be finite. Well, again you're not telling Zeno anything he didn't know already: if he started out with dividing up a finite distance into infinitely many smaller ones, then he understood that the sum of those smaller sections would add up to the original finite amount! No calculus needed here, by the way. Oh, and all philosophers will agree to this. OK, but then why do so many mathematicians point to this fact and proudly pronounce the paradox to be solved? I think I know why: it is because so often the reasoning in Zeno's paradox is misrepresented. To be specific, in popular discussions of Zeno's paradoxes (not the professional ones mind you, they know better; but the popular ones, like on many websites, espcially maintained by math enthusiasts) the end of Zeno's argument is often said to be something along the lines of: "and therefore, since Achilles always needs to make up yet another distance, it will take him an infinite amount of time to pass the Tortoise". Wrong of course, but wrong for two different reasons. "Wrong", the mathematician will say, since one can add up an infinite number of time intervals and end up with a finite amount. And, the philosophers will gree: if *that* was what Zeno said, then he clearly made a mistake in his reasoning, and pointing that out would indeed resolve the paradox. But, that's not what Zeno said. Therefore: "Wrong", the philosopher will say, because this misrepresents Zeno's argument. The right ending should instead be: "and since Achilles always has some other thing to do before he can even catch up to the Tortoise, he can't pass the Tortoise". Plain and simple. No considerations as to amounts of time this would take, but a consideration of the very notion of infinity itself: how can one finish an infinite sequence?! That seems like a contradiction in terms! How is this possible? Again, the math merely points out that there can be an infinite number of points between A and B, but that merely *restates* Zeno's *assumption* that this is the case. The real question here is about *movement*: how can one get from A to B if there are these infinitely points in between, assuming that one can't be in two different points at the same time? Or, put another way, mathematics shows us that landscapes can be dense, but philosophers like Zeno want to know how one is able to move through such a landscape! —Preceding unsigned comment added by 67.248.241.243 (talk • contribs) 15:02, 10 March 2010

Thank you, and you are correct, 67...! In a nutshell, this is the major diff between the math and the philosophy. You make it very understandable!

 —  Paine (Ellsworth's Climax)  14:36, 11 March 2010 (UTC)

Archiving

I'd like to archive everything up and to including Talk:Zeno's paradoxes#Which_country,_which_map.3F. While I'm at it, I want to set up an archive bot. Is there consensus for that? Paradoctor (talk) 15:46, 8 March 2010 (UTC)

I'd say go for it. Gabbe (talk) 19:12, 8 March 2010 (UTC)

Agree.  —  Paine (Ellsworth's Climax)  06:44, 9 March 2010 (UTC)

Fine. Paul August ☎ 02:34, 11 March 2010 (UTC)

Four yeas, no nays, that's definitive. Paradoctor (talk) 14:15, 13 March 2010 (UTC)

 Done

New Ref. section?

uhm, Paradoc?... this seems like a bit of a hazzle, since "New section" autolinks to the bottom, and new Talk contributors aren't going to know to keep this Ref. section at the bottom. How does MB deal with this on other Talk pages?
 —  Paine (Ellsworth's Climax)  00:15, 14 March 2010 (UTC)

No need to be fusssy about it, it is intended as a convenience for those who use footnotes here. Cite.php references refuse to work when the <references/> tag is placed above the footnote. Do it when you feel like it, otherwise you can safely ignore it. Who is MB? Paradoctor (talk) 00:37, 14 March 2010 (UTC)

MB is MiszaBot. I haven't seen this before, and I wondered how MiszaBot deals with it on other Talk pages. A Ref. section on a Talk page just seems odd to me for some reason. No biggee.

 —  Paine (Ellsworth's Climax)  05:07, 15 March 2010 (UTC)

Yeah, it is slightly odd. Then again, so am I, and you're in my world now. ;)

MiszaBot should have no problem, I added an HTML comment that prevents archiving. Paradoctor (talk) 09:15, 15 March 2010 (UTC)

Comment on Rucker

"However, this seems to only shift the problem: how can we distinguish in this way a moving arrow from a slightly shorter static one, just short enough to coincide with the length contraction?"[6]

That's a sensible question, can someone source it? Paradoctor (talk) 15:13, 28 March 2010 (UTC)

An object also has weight and density, which can give you a clue to what the length should be. Anyway, the comment looked like someone wanted to discuss Ruckers argument in the article rather than here. Without a source it shouldn't be included. Ansgarf (talk) 21:57, 28 March 2010 (UTC)

Is Rucker's idea discussed by any scholars at all? The biggest problem I have is with the claim that these measurements could also be done instantaneously. From a March 1 entry above, I see (mostly other) works of his mentioned & referenced, but I do not see this idea discussed anywhere at all--JimWae (talk) 22:00, 28 March 2010 (UTC)

I don't think that's what is being claimed. The claim is that if this aspect of the theory of relativity is ever confirmed, then the arrow, stopped in an instant of its travel, is instantaneously observable by the difference between its at-rest length and its contracted in-motion length.

 —  Paine (Ellsworth's Climax)  01:45, 29 March 2010 (UTC)

If I got the math right, a 100 centimetre-long arrow travelling the extraordinary speed of 1 km/s would be 99.9999999988... cm long - shortened only by 0.0000000011... cm. How could such a small difference be instantaneously observable? Using instruments to determine such a small difference would require a time interval.--JimWae (talk) 03:13, 29 March 2010 (UTC)

1 km/s is not really extraordinary. Rifle bullets feel comfortable at this speed. KE-ammo travels at 1.4-1.9 km/s, the SR-71 maxed out at 0.981 km/s, Helios II went at 70+ km/s, LEO is at 7.7 km/s (roughly that of a detonating cord), and a recent rocket sled test went up to 2.86 km/s.</ref> And that's only man-made objects.

But the real question is, how do the laws of nature distinguish between the two states of rest and motion at a given instant? For this, the magnitude of a difference is not important, only that there is one. Paradoctor (talk) 05:34, 29 March 2010 (UTC)

1 km/s is VERY extraordinary for an arrow, which is what Rucker is talking about being an instantaneously observable difference. Now if we were talking about greater speeds, there is a point at which one could observe the difference - but whether such an observation could ever be done instantaneously is another matter--JimWae (talk) 05:43, 29 March 2010 (UTC)

For starters, a strategically positioned, specially made camera would do the job, again if it were possible to make a measurement that small using the "instant" caught by the camera. It can't be impossible, not if science can devise a machine to measure the speed of light.

 —  Paine (Ellsworth's Climax)  06:27, 29 March 2010 (UTC)

1>Cameras have a shutter speed or frame-rate & 2>it takes time for light to reach the lens - observing any difference cannot be done "instantaneously" --JimWae (talk) 07:34, 29 March 2010 (UTC)

I'm not saying that it's possible yet to actually make a meaningful measurement of a moving arrow that is "frozen" in an instant in time. However, the shutter speed/frame-rate and the minuscule time it takes for reflected light from the arrow to reach the film does not negate the fact that the camera still freezes an instant in time. So if a measurement were possible, this would not only solve Zeno's paradox, but be added evidence for the reality of relativity theory, as well. Remember that Rucker's words only reflect what is true if this aspect of relativity theory is true, and Rucker makes no claim otherwise.

 —  Paine (Ellsworth's Climax)  03:29, 30 March 2010 (UTC)

• PS. According to the Length contraction article, "At a speed of 13,400,000 m/s, the length is observed to be 99.9% of the length at rest and at a speed of 42,300,000 m/s still 99%." (my emphasis on "observed") So this aspect of relativity is a reality, and Rucker's statement must be true, don't you agree? Would it be better if a reliable source for this relativistic observation were added to the claim? Or is the link to the "Length contraction" article sufficient?

There are no sources for the statement that such a length contraction has ever been observed - in fact, the length contraction article gives a source saying that this experiment has never been done. A still camera has its shutter open for more than an instant & so whatever it images is not truly an instant in time. A "video" camera has a frame rate. According to High speed photography, it seems the fastest is 600 million frames per second - with an exposure time for an image DURING a time period lasting about 1 nanosecond. I am not saying the length contraction cannot be observed (though I am surprised to see the experiment has yet to be done). I am saying the difference is not instantaneously observable, as Rucker claims--JimWae (talk) 04:30, 30 March 2010 (UTC)

I am also saying that cameras do not take "instantaneous" pictures. No matter if it's 999 sextillion frame/s or whatever - there is still an exposure time & always will be.--JimWae (talk) 04:40, 30 March 2010 (UTC)

http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html says the (not really instantaneous) photo will not show a shortening but a "rotation"--JimWae (talk) 04:51, 30 March 2010 (UTC)

Not being an expert photographer, I am just now beginning to understand. There must be a way, though. If we can stop a sine wave on the face of an oscilloscope, then we should be able to stop an arrow in flight or the equivalent. Let me think on this awhile, Jim. Meanwhile, I have transferred the claim from the Zeno's paradoxes#Proposed solutions section to the top of the Talk:Zeno's paradoxes#Rucker section above until this can be resolved, if it is indeed resolvable.

 —  Paine (Ellsworth's Climax)  09:09, 30 March 2010 (UTC)

It might be a sensible question, but I don't see where it shifts the problem. The relativistic solution does not compare an object in motion with a similar object that is motionless. It compares an object in motion with that same object when it's motionless. What intrigues me a little is the question... Would motion exist in a Universe without relativistic contraction?

 —  Paine (Ellsworth's Climax)  01:25, 29 March 2010 (UTC)

Yes. ;) And no. ^_^ Paradoctor (talk) 05:34, 29 March 2010 (UTC)

This is the article on Zeno paradox, which are still thought experiments of proverbial arrows and tortoises. These are not actual objects with a given speed, weight, air resistance, shape. BTW: You should know that any arrow on earth is moving with about the speed of light away from any observer at the other end of the universe. Speed is only relative to the position of the observer.

That said, whatever observations for actual objects have been made for length contraction, if any, this is best discussed in the article on Length Contraction. In this article on Zeno's Paradox it should suffice to note that Rucker pointed out that, assuming length contraction as defined in relativity, an arrow in motion will be different in length from an arrow in motion. If we want to include the Rucker quote at all. Ansgarf (talk) 09:18, 18 April 2010 (UTC)

Cringe-worthy section needs fixing

The section "The paradoxes in modern times" reads like someone has an axe to grind.

There are lots of references to "philosophers" (as if they were a unified team) rejecting the conclusions of "mathematicians" (again, far too simple an image). But I don't see a single source showing a mathematician claiming that Zeno's paradoxes have been resolved by mathematics! Do these misguided mathematicians actually exist? There is a feeling of "victim complex" about the whole paragraph.

I think the section should stay, but simply pointing out that the mathematics does not touch on the philosophical issues involved, rather than reading like the history of the Mathematician-Philosopher Wars. Is anyone actually disputing this point?

Alternately, if someone can find a source with a mathematician claiming that the paradoxes have been done away with by mathematics, then let's see it! And if you claim that [22] is just that, then maybe you should read the first paragraph of it. [22] is a PHILOSOPHICAL attempt at resolution, and states clearly that the philosophers are right to say that mathematics doesn't resolve the paradoxes.

121.98.145.129 (talk) 11:26, 25 April 2010 (UTC)

The mathematician/philosopher dichotomy is real. For example, this from The History of Mathematics by Burton. "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.' " To a mathematician, it seems clear that Zeno is making the false assumption that a sum of infinitely many terms must be infinite.

Some modern philosophers, on the other hand, find in Zeno an anticipation of quantum mechanics. In mathematical terms, they think that Zeno had the idea that space was discrete rather than continuous.

The article needs to represent both views.

Rick Norwood (talk) 12:27, 25 April 2010 (UTC)

As far as I understand it, the usual philosophical worry about the "mathematical" solution to Zeno's paradoxes (and particularly to the Achilles and the tortoise) is that although we have a formal system in which the sum of an infinite series is itself finite, it is not clear how or why this would apply to space and time. This need not involve anything as specific as quantum mechanics or space's being discrete.

All the best. –Syncategoremata (talk) 12:50, 25 April 2010 (UTC)

Thank you for the quick replies. But my two points still stand:

(1.) Shouldn't the paragraph include a source showing that some mathematicians think that mathematics has done away with Zeno's paradoxes? Your History of Mathematics quote would be good, although I still think the quote you showed is a little bit ambivalent. If the quote included the sentence you wrote after it, Rick, then that would be more like it! :)

(2.) This is subjective of course, but the style of the paragraph really comes across as non-encyclopedic to me. It stands out from the rest of the article and has a sort of "soapbox" quality to it, which is made worse by the four-references-in-a-row thing that you usually find in articles where someone has an axe to grind. If you don't agree with this, then of course it's just a matter of opinion, but I think a lot of people would agree with me on this. 121.98.145.129 (talk) 13:22, 25 April 2010 (UTC)

I'll add the quote above and also one by Bertrand Russell, and then a quote or two on the other side. Rick Norwood (talk) 13:24, 26 April 2010 (UTC)

A personal side note

I would like to state, quite frankly, that in order to construct the tortoise paradox, you must first complete an infinite number of steps to divide everything into half-distance slices. Therefore the very act of splitting time and space in the way they are split in the tortoise experiment requires accepting that an infinite number of tasks can be completed. This solves the tortoise paradox, by simply pointing out that the paradox uses circular logic.

Also infinitesimals solve the arrow paradox by having infinitely small slices that are not points, but rather arbitrarily small open sets centered around a point. Points are closed sets, you see, and an infinitely small open set behaves somewhat differently. Motion can occur in an infinity small open set. This solves the Arrow paradox due to the way Topology works on a dense set like the Real Numbers. (We live in a real valued plane. If we didn't, the world wouldn't work, because we would not live in a connected space.)

There. Solved. Can we stop arguing about it, and just present it the way the relevant literature does? Which is to say "Solved in the minds of mathematicians, but still argued about by philosophers due to their rejection of precision in language?"

Sorry, I don't like philosophers much. —Preceding unsigned comment added by 66.133.78.56 (talk • contribs) 03:53, 28 April 2010 (UTC)

It would be helpful if you could propose specific changes and had reliable sources for them. Paradoctor (talk) 04:55, 28 April 2010 (UTC)

I'm not really trying to get into fixing this article, but I found the page's "proposed solutions" to be unintelligible. This article got me to understand the solution, in that taking the limits will give the distance at which the tortoise is passed. Particularly assignment 4. http://web.gnowledge.org/episteme3/pro_pdfs/28-broni-prabhu.pdf Jabberwockgee (talk) 03:34, 2 May 2010 (UTC)

It is not circular logic. In the Tortoise paradox one doesn't go through an actual process of first dividing some finite interval into infinitely many smaller ones. Rather, one simply points out *that* (on the assumption that spcae is dense) there are infinitely such intervals, or points, to go through.

The article as of June 5, 2010

This article looks pretty good to me. I recall looking at it a few months ago and feeling dissatisfied. I don't know whether this is due to actual changes in the article, or my own state of mind.

Here are some comments:

I think the inclusion of Aristotle's language is a great idea. It gives a valuable background to our more modern and probably more precise formulations and also serves to alert us when our modern formulations stray too far from the original intent.

I am also grateful for the inclusion of Peter Lynds' perspective. I believe this point of view is crucial for a modern understanding. The theory of nilpotent infinitesimals, discussed in the Wikipedia article on Smooth infinitesimal analysis, is a detailed exposition of this viewpoint. I have been unable to determine the extent to which Lynds is familiar with that theory or whether he understands that this viewpoint requires intuitionistic logic.

The article would be improved by the inclusion of a brief discussion of the bearing of the theory of nilpotent infinitesimals and Smooth infinitesimal analysis on Zeno's paradoxes.

Dagme (talk) 04:02, 6 June 2010 (UTC)

Writings about Zeno’s paradoxes

The section with that name contains a list of philosophical references that I think are well justified and coherent with the topic of this article, so that I think the template call

is not justified by the content. It is also malplaced, because the section is not about influence on popular culture, it is about philosophical corollaries. I vindicate removal. Rursus dixit. (^mbork³!) 09:14, 12 June 2010 (UTC)

Convergent/divergent and harmonic series

Sorry, I just noticed the prior back-and-forth over this text... To me, the text seems overly repetitious—too much like recitals are obligatory. I don't really see how Achilles wouldn't catch up with a tortoise between 1/4 and 1/5 but how 1 meter and 1 second might help to justify that, or not ?? Over-specifying should be avoided, as should anachronism (and quantum physics, generally). It just seems increasingly off-topic to give Zeno an alternative paradox and to dwell on it until Achilles' movements become unnatural, (languishing for eternity in slow-motion baby steps or something). Regarding the interpretation section, the literal pause for thought really is a great gag, but I agree OR should be removed and that Aristotle (at least) was more interested in Zeno's countably infinite duration that comes to pass.—Machine Elf 1735 (talk) 01:06, 27 October 2010 (UTC)

So, I take it we can allow the reader to calculate that it takes 1s to travel 1m at 1m/s? I'll remove those two paragraphs and the unsourced quantum mechanics paragraph. Thanks.—Machine Elf 1735 (talk) 09:09, 29 October 2010 (UTC)

BTW, I've left notes at User talk:82.6.251.90, User talk:Raiden09 and User talk:Raiden10.—Machine Elf 1735 (talk) 21:29, 29 October 2010 (UTC)

Reply seems to miss the point completely

From the article under the section The Paradoxes in Modern Times:

"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–that both the distance and the time to be travelled are infinite. However, Zeno's problem was not with finding the sum of an infinite sequence, but rather with finishing an infinite number of tasks: how can one ever get from A to B, if an infinite number of 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][23] The Aristotle/Archimedes reply (although possibly dated by modern physics) to such an objection goes rather along the same lines as the "classical" solution to the paradox. An infinite number of events/occurrences can occur consecutively in a finite amount of time, provided the amounts of time needed for each occurrence to transpire satisfy certain constraints.[vague]"

This "Artistotle/Archimedes reply" in the second half completely misses the point of the first half, and in fact commits the very mistake that the first half was warning about! To be specific: in the paragraph above it is claimed that Zeno wasn't saying that it takes an infinite amount of time to do any kind of movement, but rather that Zeno was simply concerned with how any event can take place if infinitely many events have to (successively) precede it. In other words, any consideration of how much time it supposedly takes to do something is not the issue raised by Zeno. And yet what does this "Aristotle/Archimedes reply" say? That it only takes a finite amount of time to do infinitely many tasks! Can we please remove this "Aristotle/Archimedes reply" from this paragraph? Thanks! Bram28 (talk) 23:10, 5 November 2010 (UTC)

But then, where is the paradox? "Being concerned" is not a paradox.

Why the hell shouldn't one use time (or a simple model thereof) in order to explain something? "Precede" is a fundamentally temporal notion, so you can't say it has nothing to do with time. Water has many strange properties, for instance, being a liquid at room temperature. Suppose you observe such a property, and after doing some thinking you consider it to be strange. You call it a paradox. Suppose someone explains it to you using the concept of hydrogen bonding. Suppose you then say "ah, but I never said anything about hydrogen bonding, so you aren't allowed to either". That's stupid. There are no rules as to what is allowed to be used in a scientific explanation, so long as it stands up to scrutiny.

Saying "you can't talk about that because Zeno didn't" is stupid and irrelevant. The first humans who observed the orbits of the planets didn't talk about gravity. But the concept of gravity turned out to be quite useful.

Zeno's paradoxes simply aren't a problem anymore. Of course if you want them to be a problem, they will be, but otherwise there isn't really much to them. It was groundbreaking 2000 years ago, like a lot of things. Now we have moved on. If you want to pretend otherwise, that's your choice. Raiden10

Also what is wrong with explaining Archimedes' explanation of why Achilles can catch the tortoise? Is that taboo or something? I don't see a link to it, only a link to geometric series, a purely mathematical concept that at first blush has nothing to do with Zeno's paradoxes. At least a little explanation is called for, especially when it isn't hard. I also think it's worth explaining that Zeno closing in on the tortoise at a positive speed for all eternity does not entail that he actually catches it. Others are right when they say the apparent "paradox" results from the confusion that comes from misusing the language of discrete intervals (i.e. "first/last events", "steps" and such) to talk about what the continuous intervals by which time is modelled. The best way to proceed is to explain this confusion, introduce the proper language of continuous intervals, and from there explain how everything works smoothly in Zeno's case (and adding some strange examples of Achilles never catching the tortoise related to the harmonic series and the digamma function, which serve as a surprising and illuminating contrast to the "usual" situation). Raiden10

• "...At least a little explanation is called for, especially when it isn't hard."

  Or rather, when it is hard...

• "...worth explaining that Zeno closing in on the tortoise at a positive speed for all eternity..."

  Achilles. We're all welcome to our opinions, but as you say: "Others are right when they say..." or rather, explain—just not by introducing a "proper language", see WP:JARGON, or by elaborating on "a surprising and illuminating contrast" (for some readers), see WP:OR and WP:SYN.

• Posting on the talk page is a step in the right direction. You're welcome to be persuasive in an effort to achieve a consensus that's more in line with your personal opinions... here's some free advice:

1. Your use of multiple accounts is prohibited, (WP:SOCKPUPPET). You should pick one account and ask an administrator how to formally disclose the other accounts you've used. Failure to do so can result in all known accounts being blocked, indefinitely (and possibly your IP address range for some time as well).
2. Do not revert over and over again without establishing consensus on the talk page, see WP:BRD, and always include an WP:EDSUM.
3. It would be stupid and irrelevant to call another editor or their actions "stupid and irrelevant." Polite arguments are generally much more persuasive than confrontational ones and other editors might interpret such a comment in the context of WP:CIVIL, if not WP:AGF.
4. Always sign your posts to a talk page using four tildes (~~~~). The date, time and your customizable signature will be substituted, automatically, when the page is saved.—Machine Elf 1735 (talk) 06:26, 26 November 2010 (UTC)

So do you agree there is some need to explain the Archimedes solution to Zeno's paradox of Achilles and the tortoise (the version of the paradox which is relayed by Aristotle)? Currently all that happens is the article points to a pure mathematics page on infinite series, and then declares "these methods allow the construction of solutions". That's it. These are the best known responses to Zeno's paradox(es) to the general public, but currently Peter Lynds gets more airtime. It really wouldn't need that much, in a page that is not presently that long. (Raiden10 (talk) 19:29, 26 November 2010 (UTC))

• It would be stupid and irrelevant to call another editor or their actions "stupid and irrelevant."

  That makes two of us, then.

  (Raiden10 (talk) 19:33, 26 November 2010 (UTC))

Achilles and the Tortoise

"Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise." The conclusion drawn from this doesn't seem correct. Yes, it will take an infinite number of points (or infinite number of time intervals), but distances between the points (or the difference between the intervals) approach 0 infinitely. We assume that space and time are continuous (which seems to be supported by mainstream science), so the same can be said of any action: any action that requires an amount of time takes an infinite amount of intervals of time to complete, and any object can be divided into an infinite number of distances. This isn't so much a paradox as it is a reflection on the nature of space and time.

To continue the line of thought: if the Tortoise gets a 100 m head start and Achilles and the Tortoise move 100 m and 10 m, respectively, during, say, 10 seconds. Their speeds are 10 m/s and 1 m/s, respectively. The time it would take for Achilles to overtake the Tortoise would be given by:

(vT)t = (vA)t + d
(10 m/s)t = (1 m/s)t + 100 m
(9 m/s)t = 100 m
t = 11.11...1 s

The asymptote in this case is 11.11...1 s and 111.11...1 m. The sum of the intervals of time and the sum of the changes in distance will approach these values infinitely. MichaelExe (talk) 12:21, 16 November 2010 (UTC)

I agree that the conclusion certainly seems incorrect. In fact, my guess is that the conclusion really is incorrect, for if you let Achilles and the Tortoise race, guess what, Achilles will pass the Tortoise no problem. But, that simple observation is exactly what makes this a paradox! That is, Zeno gave an argument that concludes that Achilles cannot pass the Tortoise, and yet we all strongly believe (based on everyday observation) that Achilles will pass the Tortoise. To resolve this paradox, you therefore need to either say that our observations are somehow all illusory, and that things don't move, or you have to point out a flaw in the argument. My money is on there being a flaw in the argument. But, and here is the thing that most people unfortunately do not grasp: to show that some argument is incorrect, you have exactly two options: either you show that the logic of the argument is incorrect, or you show that one or more of the assumptions of the argument is incorrect. What is not an option, is to show that the conclusion of the argument is incorrect. Think about it: what if person A gives an argument with conclusion 'X', and person B replies to that argument by giving an alternative argument with conclusion 'not X': did person B in any way refute person A's argument? No, because person C looks at this and says: OK, so now we have 2 arguments: one that concludes 'X' and one that concludes 'not X'. Which one should I believe? And of course the answer is that there is no preference of the one argument over the other. Unless, of course, you show where either of the arguments go wrong, i.e. (again) you either show the actual reasoning to be incorrect, or one or more of the assumptions to be incorrect. Now, what you just did, was basically make an argument, based on mathematics, that Achilles will pass the Tortoise. So, basically, Zeno was person A, and you responded like person B. This doesn't help. In fact, it only strengthens the paradox: now we have two arguments with opposite conclusions. What you need to do, is to show where Zeno goes wrong in his argument. Is his logic mistaken? Does he somewhere make a bad assumption? You haven't shown either of those. All you did, as you yourself stated, is to show that the conclusion is mistaken. Also, please consider this: there are a number of questions related to the Achilles and the Tortoise race: 1. Will Achilles overtake the Tortoise? If so: 2. Where does Achilles overtake the Tortoise? and 3. When does Achilles overtake the Tortoise? Mathematical analysis can, (like you said, assuming space and time to be continuous) answer questions 2 and 3. But, that is all assuming that Achilles will overtake the Tortoise in the first place. And, Zeno argued that Achilles will not, because he can not. Indeed, Zeno was concerned with question 1, not with questions 2 and 3. Now, I know what you're thinking: how can Achilles not pass the Tortoise when I can calculate exactly when and where Achilles overtakes the Tortoise? Well, that is a good question, but that's what makes this a paradox! As long as you don't show what is wrong with Zeno's argument, you will not have resolved this paradox.Bram28 (talk) 14:54, 16 November 2010 (UTC)

The premise seems to be that it's impossible to move an infinite amount of intervals of distance in a finite amount of time. There isn't any support for this premise, and, assuming time is continuous, this premise is incorrect. Everything that happens within a certain amount of time happens through an infinite amount of intervals of time, because you can divide an amount of time any way you'd like. Every movement limited to a single finite interval of time can be divided into an infinite amount of intervals of distance or displacement. I meant to do a table, but I didn't have time (no pun intended) this morning. Here's another mathematical analysis, with the same speeds and head start for the tortoise, which permits a better visualization of the problem:

Achilles' distance, dA, m	Tortoises's distance, dT, m	variation in time, Δt, s
0 m	100 m	0 s
100 m	110 m	10 s
110 m	111 m	1 s
111 m	111.1 m	0.1 s
111.1 m	111.11 m	0.01 s
111.11 m	111.111 m	0.001 s

The sum of the time intervals (using only the values in the table) is given by:
10+1+0.1+0.01+0.001 = 11.111 s
This, times the speed of Achilles gives his distance (11.111 x 10 = 111.11)
And, if you take the total difference in distance of the tortoise (111.111-100=11.111 m), and divide that by the time (11.111 s) and you get 1 m/s, the speed of the turtle.
So, in the table, you see that the total time it takes would approach 11.11...1 s, infinitely, but what happens after 12 seconds, or even 11.11...12 seconds? Achilles passes the Tortoise. Zeno basically assumed that time stops at 11.11...1 s; nothing can happen at the end of the interval. I would say Zeno's mistake was his premise. Basically, an infinite amount of intervals of time =/= an infinite amount of time (and an infinite amount of intervals of distance =/= an infinitely long distance), because we see that these intervals of time get smaller and smaller. It's like how a perfect circle would have an infinite amount of symmetries (reflections through its centre or rotations). Does this fact make the circle infinitely large? No. The infinite amount of time would be how long it would take to calculate the amount of time it takes Achilles to overtake the Tortoise, exactly (with decimals), because the number will never end. XP

In "The dichotomy paradox", it states "This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility." Zeno hasn't (as far as I can tell) explained why this is impossible. It also states "This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all." This, I'd say, is correct (if space is continuous), but the first point/distance is necessarily arbitrary, so I fail to see any paradox in this.

In "The arrow paradox", it states "In other words, in any instant of time there is no motion occurring, because an instant is a snapshot. Therefore, if it cannot move in a single instant it cannot move in any instant, making any motion impossible." In continuous time and space, there's no such thing as an instant, so again, the premise is flawed. A snapshot takes time. If, somehow, we can prove that there are instants of time (the chronon, possibly Planck time), then the paradox would be valid. But, so far, we have no credible evidence of the existence of quantized time (and, it seems, we don't understand the significance of Planck length that well). MichaelExe (talk) 21:52, 16 November 2010 (UTC)

Sorry, but Zeno does not use the premise that it takes an infinite amount of time for an infinite number of events to (sequentially) take place. This is a common charge against him, but unfortunately it misrepresents his argument (this is also stated in the main article). And, it really just misses the all important point, which is that it is the very nature of infinity that does not allow for any infinite sequence to ever be completed. The amount of time (whether finite or infinite) is not a consideration in his argument; it is that there is always (at least) one more event to take place before any such sequence can be completed. For Achilles to overtake the tortoise, an infinite sequence, i.e. a sequence without end or finish, needs to be ended or finished, which is a contradiction in terms. Time is not brought in by Zeno. It is the later mathematicians that somehow felt that Zeno, and somehow all other Greeks, had no understanding of the nature of infinity, and thought that the sum of an infinite number of terms somehow must be infinite. But that is just a straw man, and frankly a ridiculous charge: Zeno starts out by infinitely dividing a finite interval in half, from which of course it then immediately follows that the sum of those intervals is the original finite interval. Are we really to believe that Zeno and all Greeks were so stupid not to realize that? No! In fact, it is the very mathematical fact that in a continuous space/time geometry any finite interval contains infinitely many smaller intervals that is the very *starting* point for Zeno's argument. And, what Zeno wanted to know is: how is movement through such a geometry at all possible? Again, mathematics doesn't solve anything here, but instead only strengthens the paradox: mathematics shows that space/time can be such that between any points in space/time, no matter how close apart, there are infinitely many other points. So, what Zeno then asks us is: how is movement through such a landscape possible? 67.248.246.8 (talk) 03:54, 17 November 2010 (UTC)

The "premise" is that it is impossible to complete a task that has no final step (think of recursion) -- and, in the case of the dichotomy, also no first step. Whether time "is" continuous or discrete does not affect that premise. Anyway, besides what we may or may not be able to "prove" here, there is no consensus among scholars that the paradoxes have been resolved. --JimWae (talk) 23:11, 16 November 2010 (UTC)

That premise could then be extended to any action that occurs over a period of time. And, because we (and Zeno) treat space much like we treat time, it would mean any length has neither a beginning nor an end. The first and last intervals could always be divided further. Together, this would essentially imply that continuous time and space, in-and-of themselves, are paradoxical, because no amount of time, nor any length, could be complete. At this point, it may be more a question of what it means to divide by infinity than anything else, because you simply can't divide by infinity and expect a number that has an end (although 0.999... is considered equal to 1, but I don't know if you can extend this to 0.000...1 = 0). Dividing anything by infinity simply leaves no beginning nor end. If time (or space) were discrete, however, there would always be a first and a final step, because there would be a indivisible amount of time (or space) that could start or end, because you couldn't divide it further. The premise you highlighted would still stand; it just wouldn't be relevant. MichaelExe (talk) 02:22, 17 November 2010 (UTC)

I just realized that my arguments are pretty well covered in the Proposed Solutions section, by Saint Thomas Aquinas and Peter Lynds (whose work on space and time actually lead me back to Zeno's paradoxes, in the first place, even though I'd heard of Achilles and the Tortoise). MichaelExe (talk) 03:16, 17 November 2010 (UTC)

And even though there may not be a first or last step, the sum of the steps still has a beginning and end (i.e. at 0 m and 0 s, and at another time and distance based on speed). MichaelExe (talk) 12:00, 17 November 2010 (UTC)

Which is just a perfect demonstration of the fact that we're talking about different issues, and trying to solve different problems. You are talking about the nature, or static gemometry, of the time/space landscape, about where and when things are happening. Zeno was talking about the dynamics, about the movement through such a landscape. So to you, the presence or absence of any first or last step is indeed irrelevant. To Zeno, it is vital. Bram28 (talk) 14:14, 17 November 2010 (UTC)

Movement is just a variation in both the space and time landscapes. The first point is (0,0) (for time and distance) and the last is also finite. I don't see how it really changes anything if we isolate time and space from each other. It's still logical that there would be no first or last interval (or step) in either space or time, because they could always be further divided. The implication is that in continuous time and space (where there is an infinite amount of intervals or steps), there is no first or last step, but there are beginning and ending points. Why would having no first or last step even imply that motion is a paradox or illusory? It may be common sense that you cannot complete an action without a first or last step, but common sense would be wrong, because when we think of steps, we tend to think of discrete, indivisible steps. The problem arises in translating the logic of discrete intervals to that of continuous intervals, which isn't logical. Zeno applied the logic of discrete steps (where there's always a first and a last one) to continuous steps (where there's never a first or a last one, but still a beginning and end). MichaelExe (talk) 20:53, 17 November 2010 (UTC)

the race is logically absurd

For the race defines the finishing point of achilles as the start point of the tortoise. So its by definition impossible to overrun. —Preceding unsigned comment added by 125.18.235.210 (talk) 07:20, 19 December 2010 (UTC)

No, read it again. The goal is to catch the tortoise. --JimWae (talk) 07:34, 19 December 2010 (UTC)