Talk:Zeno's paradoxes/Archive 5

Mediation
Can I firstly apologise on behalf of MEDCOM for no mediator taking this case on. If you still need a mediator, I will happily take it on as I am available to mediate. Seddon talk|WikimediaUK 07:01, 5 January 2010 (UTC)


 * This is not the appropriate section for mediation. The mediation page is at http://en.wikipedia.org/wiki/Wikipedia:Requests_for_mediation/Zeno%27s_paradoxes
 * Steaphen (talk) 13:19, 15 January 2010 (UTC)
 * I know :) but since im on medcom, I'm asking whether there is still a need for a mediator :) Seddon talk|WikimediaUK 04:30, 31 January 2010 (UTC)
 * The mediation issues in the main have not been resolved. If you would like to make relevant representations on the appropriate pages, that would be appreciated. I expect you've noted that the POV notice in the main article was, in violation of Wikipedian guidelines, removed without the issues being resolved. Hence the likely need for arbitration, given the persistence of bias and POV in the main article.Steaphen (talk) 08:48, 1 February 2010 (UTC)

Arbitration called
Arbitration/Requests/Case Steaphen 13:37, 11 February 2010


 * I find the use of "called" (instead of "requested") ambiguous and confusing. Please cite reliable sources for this use of the word.--JimWae (talk) 20:35, 11 February 2010 (UTC)


 * Dear JimWae, "called", "requested" ... take you pick. It is irrelevant.


 * This arbitration is not about content! It is about inappropriate behaviour, of making statements that are not supported by Reliable Sources, and are thus speculative opinions and assumptions. Plain and simple.


 * If any competent physicist can assert that via ANY mathematical means, we may fully and precisely account for physical movement of physical objects (no matter what their size), I would like to see that statement presented here for the benefit of this arbitration.


 * Any by "fully account" that means experimentally and theoretically supported by the evidence, and the quantum theory, or whichever theory has peer-reviewed support, and accounts for the experimental data.Steaphen (talk) 00:57, 12 February 2010 (UTC)

Outside opinion by CBM
The underlying dispute is something that mathematicians can sometimes be insensitive to. It's certainly true that, in the standard Newtonian mathematical model of motion, Zeno's paradoxes are no issue. From the viewpoint of a mathematician, this is all that matters. Thus many calculus textbooks say that Zeno's paradoxes have been solved, because from a mathematical standpoint they have been.

However, this isn't a very pleasing answer for non-mathematician philosophers and physicists. Also, the general naiveté with which Zeno presented the paradoxes makes it difficult to tell what the paradoxes actually are. In our mathematical reductionism we can easily take them to be statements about Newtonian mechanics, while physicists might take them to be referring to actual motion rather than to our model of it. The same situation arises very often, when mathematicians approach a vaguely-worded philosophical problem by first making it mathematically precise and then solving the precise version as if it was the same as the original vague version.

In the article here, there is presumably space to cover both the mathematical solution and the more general philosophical discussion. I did a Google search earlier and it looks like there is a decent philosophical literature on the subject, which isn't surprising. I don't know if there is literature that explores the relationship between quantum physics and Zeno's paradoxes.

My general advice, as an outside observer, is that it might be best to discuss the mathematical solution in its own section, making a note that it relies on the Newtonian model of movement. Within that section, things like quantum physics are irrelevant. This is the solution that is commonly presented in calculus textbooks.

In a separate section, the article could discuss more general philosophical research on the paradoxes. The point here is not that the Newtonian solution is invalid, but that there may be other concerns that are not captured by the Newtonian model.

Of course these two approaches (mathematical/philosophical) are not in conflict. They complement each other by revealing different aspects of the situation. &mdash; Carl (CBM · talk) 17:52, 11 February 2010 (UTC)


 * "the general naiveté with which Zeno presented the paradoxes": I'll bet you a penny that you can't cite that to reliable sources. Paradoctor (talk) 18:22, 11 February 2010 (UTC)


 * A google books search for "zeno naive paradox" will find several interesting examples, but maybe not in the sense of "naive" that I had in mind. &mdash; Carl (CBM · talk) 20:54, 11 February 2010 (UTC)


 * Keep looking, that penny won't rust away. You might save yourself some work, though. Ask yourself: What would it take to make your statement true? Paradoctor (talk) 21:28, 11 February 2010 (UTC)


 * My opinion is quite simple: the paradoxes as originally stated refer to our naive conceptions of position, time, and motion, rather to any particular formalism in which they could be either proved or refuted. If we disagree on that, I don't think it's worth discussing in great depth. &mdash; Carl (CBM · talk) 21:55, 11 February 2010 (UTC)


 * "the paradoxes as originally stated": Stated where? Paradoctor (talk) 22:24, 11 February 2010 (UTC)


 * I agree with the thrust of CBM's comment, but with regards to "naiveté" consider the following:
 * In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance... Bertrand Russell, The Principles of Mathematics (1903)
 * Paul August &#9742; 18:58, 11 February 2010 (UTC)


 * "two thousand years of continual refutation": I presume Mr. Russell has provided either citations to literature reviewing the reception history of Zeno's paradoxes, or a review of his own? Paradoctor (talk) 19:20, 11 February 2010 (UTC)


 * Doesn't that published quote constitute "a review of his own"? &mdash; Carl (CBM · talk) 20:10, 11 February 2010 (UTC)


 * Not in any understanding of the term "review" I know of, and most assuredly not in the specific meaning alluded to, I'm afraid. The quote might be considered a short statement of the conclusions drawn from such a review. Without supporting citations or argument, this is "just" Russell's opinion. Considering Russell's statement about the history of the topic, one should be able to find a lot of reliable sources contradicting Russell, which might lead an adventurous soul to worry about WP:UNDUE should those voices not get mentioned. Paradoctor (talk) 20:30, 11 February 2010 (UTC)


 * Thanks to Carl for making the discussion more promising. I always believed (naively, or not?) that Zeno hinted at something like that: it is unbelievable that mathematical models stipulating infinitely many points in a finite domain reflect the reality in this aspect (but of course they are quite good in many other aspects). (I never read something deep about Zeno paradoxes.) I wonder, do you agree or disagree? (Sorry if it is off-topic.) Boris Tsirelson (talk) 21:42, 11 February 2010 (UTC)

I apologize; I was not trying to start a long discussion about Zeno's paradoxes. I just wanted to point out that there is room to discuss both the formal mathematical solution (although it may not solve the original problem) and philosophical aspects of the original problem (although these may be philosophical, rather than mathematical). We don't have to choose between these alternatives, because they actually reinforce each other. &mdash; Carl (CBM · talk) 21:58, 11 February 2010 (UTC)


 * I agree, completely. Boris Tsirelson (talk) 22:16, 11 February 2010 (UTC)
 * Just as a reminder. The current article does mention the philosophical aspects as well as the mathematical aspects of the original paradox, and that more than once. It also mentions that when applied to physical reality quantum aspects may play a role. These paragraphs may be improved, but it is not the case that the current article doesn't mention it. It does.Ansgarf (talk) 23:27, 11 February 2010 (UTC)
 * Yes, you're right. I read through the comments higher on the page, and I thought that both sides were right, but about different things. The lede section of the article has a nice tone, I think. The difficulty is the section "Status of the paradoxes today" lower down, which tries to do too many things at once. I would propose adding a section "mathematical resolution" that discusses the calculus-textbook approach, and then editing "status of the paradoxes" to focus more on the philosophical reception of the mathematical solution. Looking through google books, I am sure there are enough sources to do all of this. &mdash; Carl (CBM · talk) 00:51, 12 February 2010 (UTC)
 * A textbook solution just got removed a few weeks ago, since too many people felt that the paradox is at its core not about calculus or algebra. In my opinion it wasn't worth the trouble to be included . But if you can come up with a paragraph that that avoids the problems the old version had, I wouldn't object.
 * Mentioning Brouwer at that point was a compromise, on an earlier formulation. It claimed that Intuitionists reject to any use of infinites, which is not true. . The most recent sentence on Brouwer was intended to keep those people happy, while at the same time keeping them from adding false claims about Intuitionists. I agree that the whole topic is a bit spurious, but it is a compromise. Ansgarf (talk) 01:14, 12 February 2010 (UTC)
 * I see now. Regarding the calculus solution: to many mathematicians, including me, the calculus solution "is" the solution of the paradoxes. It's certainly repeated as a solution in numerous calculus and analysis texts. Moreover, I saw several sources today that seemed interested in discussing whether it really was a solution. I think the article would be incomplete without covering the mathematical viewpoint. On the other hand, I understand the philosophical viewpoint that the mathematical solution is too idealized to address the original problem.


 * Regarding Goedel and Brouwer, as a logician I think discussing them at all is a red herring. The issues that the intuitionists have with infinity (and you are right that they do not reject it outright) are not related to infinite sums nor to classical mechanics, while Zeno's paradoxes are unrelated to the law of the excluded middle. Goedel's theorem is not related to the foundations of calculus, which were accomplished in the 19th century anyway, well before his time. &mdash; Carl (CBM · talk) 01:42, 12 February 2010 (UTC)
 * Ok, maybe there is something to be said to leave out everything that is too tangential. And Intuitionism probably is. If you can include a short treatment of the mathematics of the paradox - which might already be achieved by reorganising the current article a bit - then I'd be more than happy. The article does are ready contain references that calculus is not "the" solution to all aspects. By some reorganisation this might become more pronounced. Ansgarf (talk) 02:30, 12 February 2010 (UTC)

[From Steaphen] - The arbitration was called on the issue of inappropriate behaviour -- regarding statements being made that are not, and cannot be supported by real-world evidence, and thus remain in the domain of speculation. Zeno's Paradoxes concern the subject of movement of physical things -- runners, arrows etc. If mathematics can assist in that inquiry, well and good. If spirit-guides of recently deceased can assist in that inquiry, well and good. If a bumbling idiot can assist in that inquiry, well and good. Whether mathematics can assist in solving Zeno's Paradoxes is not relevant or useful until a competent physicist can detail how theory matches and accounts for reality (experimental data concerning the movement of physical things).Steaphen (talk) 01:20, 12 February 2010 (UTC)


 * As I was saying, there are two sides to resolving the paradox: the mathematical resolution within Newtonian classical mechanics, and the philosophical discussion about the real world. Each of these is, in its way, important to a complete understanding of the paradoxes. &mdash; Carl (CBM · talk) 01:42, 12 February 2010 (UTC)


 * [From Steaphen] - Upon what basis do you affirm that mathematics is important to this issue, if there is no reliable linkage, correspondence or congruency of theory with evidence and fact? A lack of correspondence (of theory with fact) is how people get burned at stakes. Please start with observable reality, and wind backwards into theory (as we might expect of any good scientist or serious thinker) -- if that theory involves mathematics, well and good. If not, so be it. Starting with observable reality is I believe valid and worthy. Let's leave speculation and baseless opinions for forums devoted to such things.Steaphen (talk) 02:06, 12 February 2010 (UTC)


 * I think that mathematics is important to the issue because large numbers of mathematics books bring up the issue in the context of infinite series. Most mathematicians learn of Zeno's paradoxes in this way. Moreover, I was looking at philosophical references today and many of them mention the mathematical solution (in some cases, only to criticize it).  If you would like, I can make a list of such references. However, if you have looked into the literature on Zeno's paradoxes, I am sure you have already seen what I am talking about. &mdash; Carl (CBM · talk) 02:09, 12 February 2010 (UTC)


 * Carl, I am aware of the preponderance of opinions on this issue. Arbitration was called to seek some clarity amongst the clamour and noise of the crowd ("large numbers"). Scientific principles ('scientific method') regarding this subject have been discarded, ignored or simply denied. My intent is to bring some discipline to the issue, by reaffirming the root validity of applying the scientific method to the subject of Zeno's Paradoxes.


 * I have not seen any examples (regarding the subject of Zeno's Paradoxes at this site) of strict adherence to one of the root principles upon which great scientists and thinkers have stood since time immemorial - the scientific method of questioning observable reality and finding a theory which fits the facts. Steaphen (talk) 02:19, 12 February 2010 (UTC)


 * I'm not sure what you're saying. The goal of this article is to summarize what is already known and written about Zeno's paradoxes – including what is written in mathematics texts. The goal here is not to solve the paradoxes ourselves (which would be unlikely) or to decide that entire branches of the literature on the paradoxes should be discarded. &mdash; Carl (CBM · talk) 02:23, 12 February 2010 (UTC)


 * I called arbitration, not because of the historical content about the paradoxes, but about claiming (or even inferring) that said beliefs, theories or literature, actually account for physical movement. There's an important difference about reporting on the literature (and theories) and claiming that the theories are valid.


 * This arbitration would not have been called IF the theories were put in their proper context -- that they remain unsubstantiated theories! and do not (at least not from the evidence I've seen) offer congruent, verifiable solutions to the paradoxes. Statements like "Using ordinary mathematics we can calculate, (or arrive) ..." are simply biased opinions with no basis in verifiable fact.


 * By all means, report on those baseless opinions, but to state that "using ordinary mathematics, we may ... " is wrong until proven and confirmed by a Reliable SourceSteaphen (talk) 02:50, 12 February 2010 (UTC)


 * Steaphan, are you even aware that the sentence you keep harping about has not been in the article since last year? Not that it was not true that "using ordinary mathematics, we can calculate a position and time at which Achilles would catch the tortoise". --JimWae (talk) 06:36, 12 February 2010 (UTC)


 * Surely, every given physical theory does not describe reality completely (and therefore will be replaced some day). Surely, this is true in particular for the concept of mathematical continuum as a model of space-time. However, all that does not mean that we should abandon all physical theories (as Steaphen seems to propose). Boris Tsirelson (talk) 07:22, 12 February 2010 (UTC)

[From Steaphen] -> From a verbatim extract of the front-page article as at 6.34 pm Australian Eastern Time, 12th February, 2010: ". While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise, ..."

Wrong. It cannot. At least not on the available evidence. Hence the request for arbitration, re the persistent violation of Wikipedia policy regarding statements without merit, that pust POV, and lack Reliable Source support.

Boris, please re-read my words. It is not about denying the history of the paradoxes, or what's been said about them, it's simply the baseless claims (see above) that are being called into question.

It's really not that hard to understand ... if you want to make claims about theories that cannot be substantiated via physical experimentation or evidence. then you must also allow other claims such as astrology, or numerology or other belief-systems that have also not been strongly correlated with the facts of reality, or that have been experimentally substantiated. Steaphen (talk) 07:42, 12 February 2010 (UTC)


 * Yes, it's really not that hard to understand: every theory "cannot be substantiated via physical experimentation or evidence", if you require it to be absolutely right. Does it mean that we should stop using theories at all, however? Boris Tsirelson (talk) 08:09, 12 February 2010 (UTC)
 * When the article talks about "While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise, ..." it refer to the tortoise and the runner in the mathematical model of the paradox. To solve the equations you are not doing "experiments", you do math. Which will give you mathematically precise solutions. That is much more rigorous than any experimentation can be. They are not any particular runners or tortoises that have a shoe size, a weight, or feeding habits. That Zeno's paradox is a thought experiment, a proof by contradiction, and not an actual physics experiment, is obvious from the context, and even mentioned explicitly in the article. Ansgarf (talk) 08:25, 12 February 2010 (UTC)


 * @Steaphan::For this discussion to go anywhere, you will need to be more "on-target" about what parts of the article you are objecting to and not throw in red-herrings. While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise could easily be changed to While mathematics can be used to calculate where and when the moving Achilles would overtake the Tortoise but I gather even that would not satisfy you. Just because we cannot specify to a million decimal points the time and position does not mean math cannot be used to calculate a time and position. Besides, Zeno's argument is that Achilles will never catch the tortoise at all, not that the position cannot be determined with infinite precision. If math can determine a time and position (even with lots of imprecision) at which Achilles has actually done more than caught but has actually passed the tortoise, math is still the tool that is being used to do so. If people do not maintain that Achilles actually catches and passes the tortoise, then they should not consider this to be a paradox at all, but some kind of "truth". --JimWae (talk) 08:44, 12 February 2010 (UTC)


 * It is not a problem to overtake the Tortoise (as Zeno surely understood). It is a problem, whether a finite time interval can contain infinitely many events, or not. Boris Tsirelson (talk) 10:05, 12 February 2010 (UTC)
 * As Carl emphasizes, we should take into account both sides of the story: (a) some models resolve the paradox admitting infinite divisibility of space-time (the mathematical side); (b) it does not mean that infinite divisibility is a property of reality (the philosophical side). Let me repeat Carl's phrase: "Of course these two approaches (mathematical/philosophical) are not in conflict. They complement each other by revealing different aspects of the situation." Boris Tsirelson (talk) 10:10, 12 February 2010 (UTC)


 * Hi, Steaphen. I've been reading the discussion with intense interest, and then I came to what seems to be the focus of your discontent.  You feel that the claim that math can be used to calculate when and where Achilles will catch the tortoise, is an unsubstantiated claim that is wrong, incorrect, without merit and so forth?  So I'm seeing this in my mind in the simplest possible, common-sense terms.  Say that Achilles gives the tortoise a lead of 1000 feet.  Now suppose the tortoise's speed is 1 foot per minute, and Achilles runs at a speed of 1000 feet per minute.  One minute after the race begins, the tortoise will have moved 1 foot, and Achilles will have moved 1000 feet.  At that point in time, Achilles would be exactly 1 foot behind the tortoise.  In the next minute, the tortoise will have covered another foot, and Achilles will have covered another 1000 feet.  So after two minutes, Achilles will then be 998 feet out in front of the tortoise.  Common sense rules that, since Achilles now leads the tortoise, then at some point he had to have passed the tortoise.  Are you then challenging whether or not math can be used to precisely determine that point in time and distance traveled, that brief instant, when Achilles and the tortoise were "neck-and-neck"? side-by-side?


 * Using the above figures, we can readily see that Achilles and the tortoise will be neck-and-neck at a time between 1 and 2 minutes, and the distance from Achilles' starting line will be between 1001 and 1002 feet. It is also certain that the time will be much closer to 1 minute than to 2 minutes, and the distance much closer to 1001 feet than to 1002 feet.


 * So how would we go about calculating the exact time and distance? Since few of us actually like math, I will not include it here save for an explanatory LINK that yields the outcome.  Using our figures, Achilles will catch up to the tortoise when he has run 1001.001001 feet.  And since his speed is 1000 feet per minute, he will be neck-and-neck with the tortoise when 1.001001001 minutes have elapsed.  So it would appear, Steaphen, that the claim is substantiated, right, correct, has merit and so forth, don't you agree?


 * (Yes, I realize that this does not even come close to doing justice to the philosophical side of this near-2500-year-old-and-still-kickin' paradox; however, it does clearly show that those seedy, unphilosophical (aphilosophical?) mathematicians are certain that Zeno's paradox has been "solved".)
 * &mdash; Paine ( Ellsworth's   Climax )  11:36, 12 February 2010 (UTC)


 * [From Steaphen] - "My discontent' is with the clearly unscientific approach to this issue. You may perform calculations, but whether they have any relationship or correlation with reality is the question. If you've observed astrologers or numerologists, they apply similar thinking and arguments to what I have seen here. Theories and calculations can be cited, but without concrete correlations with the facts, they are of equal merit. The mathematics, irrespective of however strong the illusion, appearance or approximations is irrelevant if it cannot account for the minutia of physical movement.

As far as I'm concerned, the comments here at this site are no more valid, scientific or rigorous than those of astrologers and numerologists. I'm open to theories that show congruency to the facts -- irrespective of whoever espouses them.

The mathematical arguments are, when studied in detail, irrelevant, or as relevant as astrology, in regards to the solution to Zeno's Paradoxes. Earlier I presented my understanding of the deeper nature of reality, which involves quantum superpositions of possibility and nonlocal fields of potentials, all of which will not, now or ever, be reducible to simple geometric analysis. But that is not what this arbitration is about ... it is the clear violation of Wikipedia policy of providing statements and theories that are not supported by Reliable Sources.Steaphen (talk) 12:06, 12 February 2010 (UTC)


 * But what about the solution (to the dispute, not to the paradox) proposed by Carl? Does it satisfy you? Any objections? Recall it: "Of course these two approaches (mathematical/philosophical) are not in conflict. They complement each other by revealing different aspects of the situation." OK? Or not? Boris Tsirelson (talk) 12:19, 12 February 2010 (UTC)
 * If I am not mistaken Carls proposal is to state that "(a) some models resolve the paradox admitting infinite divisibility of space-time (the mathematical side); (b) it does not mean that infinite divisibility is a property of reality". I am happy with this distinction, and I assume most people are, since this distinction is already reflected in the article, and has been in there for a long time. Ansgarf (talk) 14:07, 12 February 2010 (UTC)
 * Okay, Steaphen, I shall continue to look for reliable sources. I realize that WP sometimes frowns upon YouTube as a source, however since the content guideline clearly states that there is no blanket ban against YouTube, and since the LINK I gave above is a video made by a reputable professor of mathematics at U. of Helsinki and Florida State University, perhaps then we could begin with THIS LINK as a reliable source and an inline citation?  That web page prominently links to the YouTube video I cited above.  Please keep in mind that it is not our job to debate the TRUE vs. FALSE, the RIGHT vs. WRONG, SCIENCE vs. PSEUDOSCIENCE, etc. of any reliably sourced claim.  All we must do is agree that the source(s) is reliable.  So do you accept

THIS LINK as a reliable source for the claim: "While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise . . ."
 * &mdash; Paine ( Ellsworth's   Climax )  13:36, 12 February 2010 (UTC)

[From Steaphen] - does the Reliable Source account for/detail/explain the minutia of physical movement? No. Does astrology account for/detail/explain the minutia of physical movement? No. Then include both theories (astrology and mathematics) since "they complement each other by revealing different aspects of the situation."

Mathematics does not, in detail, determine when Archilles overtakes the tortoise. Provide one physicist who affirms that we can calculate such things, at and below the Planck length. The continued inclusion of mathematics (as offered above, and as per related links) is simply bad, incompetent science when applied to the issue of Zeno's Paradoxes.Steaphen (talk) 15:00, 12 February 2010 (UTC)


 * Please prepare yourself, Steaphen, because your argument is about to be refuted: The RS does NOT have to account for details/minutia of physical movement; it ONLY has to support the claim being made in the article.  And just because math may "complement" the pseudoscience of astrology does not detract from the solid gold fact that math ALSO complements, supports, and even validates science, as well.  Mathematics indeed DOES, in detail, determine when Achilles overtakes the tortoise, and it can do this to six decimal places for distance and nine decimal places for time.  That's pretty precise, isn't it?  That's pretty detailed.  You won't find any physicists worth their salt who will affirm that anything at all can be calculated at or below the Planck length, simply because the Planck length is, BY DEFINITION, the shortest length that "has meaning".  This in NO WAY cripples mathematics for yielding precise, detailed and PRACTICAL results ABOVE the Planck length.  And once more, it simply does NOT matter what you or I think about the science, good or bad or competent or incompetent when applied to the issue of Zeno's paradoxes.  All that matters is that the source is reliable and that it backs up the claim made in the article -- AND THAT IS ALL THAT MATTERS.


 * It is not up to me, and it is not up to you whether or not the claim is valid. The only thing that we editors get to decide is whether or not a claim can be reliably sourced.  And the claim about the mathematics that, in the eyes of mathematicians resolves Zeno's paradoxes is a valid claim and can be reliably sourced.
 * &mdash; Paine ( Ellsworth's   Climax )  15:55, 12 February 2010 (UTC)


 * PS. Would anybody else like to add their opinion about the reliable source I cited? If nobody objects and gives a reason to blackball the source, I shall add it soon to the article.
 * I don't see why we would ever cite a video in this article. But there are plenty of professionally published texts (in particular, calculus textbooks) that discuss Zeno's paradoxes from a mathematical viewpoint. &mdash; Carl (CBM · talk) 16:41, 12 February 2010 (UTC)


 * Great! Perhaps you can slip one or two of those PPTs after the claim in question?  And just to provide a focus in case it might be needed, I'll place a  template in the section of the article where the reliable source is called for by Steaphen.
 * &mdash; Paine ( Ellsworth's   Climax )  19:16, 12 February 2010 (UTC)

Steaphen: the underlying point of looking at a formalism such as Newtonian mechanics is that the formalism does not account for every possible detai. Neverthless, one can do calculations within the model to see what the model says. For example, when we want to see how high a launched projectile will fly in free fall, we don't ordinarily pull out our quantum physics textbooks. In situations like that, we just use the normal Newtonian equations to calculate it. We often ignore air resistance, too, which is much more important there than quantum effects. It seems to me that your argument would say equally well that we cannot compute how high a projectile will fly without quantum mechanics, and therefore Newtonian mechanics does not actually say how high a projectile will fly.

Similarly, if we just want to figure out when Achilles will pass some point, we can use the Newtonian model to see what it says. Of course the Newtonian model doesn't account for quantum mechanics; that's part of the point of using a model. The Netwonian model doesn't completely resolve Zeno's paradoxes, but seeing how those paradoxes play out in the Newtonian model is relevant to understanding the paradoxes, and it's also important for seeing why the Newtonian model is internally consistent. &mdash; Carl (CBM · talk) 16:41, 12 February 2010 (UTC)

Naïve question
Is this helpful? To wit, does anybody doubt that Achilles will overtake the tortoise at
 * $$t_\text{pass} = \frac{l_\text{tortoise}}{v_\text{Achilles}-v_\text{tortoise}}$$

and
 * $$l_\text{pass} = v_\text{Achilles}\cdot t_\text{pass}$$,

where $$t_\text{pass}$$ and $$l_\text{pass}$$ are the time and distance from the start when Achilles passes the tortoise? Paradoctor (talk) 20:22, 12 February 2010 (UTC)


 * I don't doubt it, Paradoctor, however when another editor sees the need for a reliable source to support a claim, then I don't see how this can be ignored. I placed the cite-needed template at the precise place in the text that follows the claim that's in dispute just in case any other editors wanted a quick focus.  Please note that I've checked the three cites that follow the next word, "Philosophers", and they do not appear to support the math claim.
 * I do not have access to, nor would I understand very well, the calculus text(s) that would make good, reliable sources, or I'd do it myself.
 * &mdash; Paine ( Ellsworth's   Climax )  20:33, 12 February 2010 (UTC)


 * 'three cites that follow the next word, "Philosophers"': They are not meant to, they relate to the following claim.
 * As expected, JimWae came up with sources supporting my contention. The challenge has been met. Or does anyone have any source proposing alternative values for $$t_\text{pass}$$ and $$l_\text{pass}$$, Steaphen? Apart from suggesting that physics is so fundamentally wrong that it overlooks the impossibility of motion, of course. Paradoctor (talk) 21:22, 12 February 2010 (UTC)


 * Well, there again, it's very easy to get off the track here and start talking about the article's content. Seems to boil down to philosophy vs. mathematics/physics (theoretical? or is that too close to philosophy?).  At any rate, reliable sources have been found, one has been chosen and added to the article, and hopefully this satisfies editor Steaphen's notable idea, and the POV maintenance tag can be dusted.
 * &mdash; Paine ( Ellsworth's   Climax )  22:50, 12 February 2010 (UTC)

Here are some text sources to choose from: --JimWae (talk) 20:38, 12 February 2010 (UTC)
 * http://nongnu.askapache.com/fhsst/Mathematics_Grade_12_Ch40_Differential_Calculus.pdf pg 510
 * http://cvs.gnowledge.org/episteme3/pro_pdfs/28-broni-prabhu.pdf p 178
 * http://books.google.ca/books?id=QsJKB4V_DmkC&pg=PA43&lpg=PA43&dq=achilles+tortoise+%2Bgraph&source=bl&ots=67boCXfnQR&sig=RK13rCkMrbm7--S1iiHd1wit1qk&hl=en&ei=d0wDS_7kE4fqsQPv_8W4BA&sa=X&oi=book_result&ct=result&resnum=8&ved=0CCEQ6AEwBw#v=onepage&q=achilles%20tortoise%20%2Bgraph&f=false
 * http://www.learner.org/courses/learningmath/algebra/session5/part_c/index.html
 * http://www.tc.umn.edu/~burc0050/math/achilles_paradox.html
 * http://doctorlaismath.com/example2


 * I think that puts you firmly within the "No" coalition. ;) Paradoctor (talk) 21:22, 12 February 2010 (UTC)


 * Yes! definitely a "No". Jim, if I had to choose from these best of the best (hard choice, truly), I'd opt for the first one.  No, No, don't sweat it, I'll add it in, and you and others can improve it if you feel the need.  Thank you very much for your effort and time!
 * &mdash; Paine ( Ellsworth's   Climax )  21:54, 12 February 2010 (UTC)

✅ – That said and done, would anyone object to adding Mika Seppala's video link (a non-YouTube version) to the External links section? &mdash; Paine ( Ellsworth's   Climax )  22:25, 12 February 2010 (UTC)


 * Add one pushover !vote. Paradoctor (talk) 22:52, 12 February 2010 (UTC)


 * Thank you all. You have clearly illustrated the process of witch-hanging, heretic-burning in detail. We may expect that onlookers at such spectacles in the past similarly remarked, "a pushover result, she didn't even try to jump on her broomstick while she hung."


 * Science and reasoning saved us from the superstitions of middle and later ages, but what or who is going to save us from the stupidity and cowardice of the contemporary dark-age of ignorance, dogma and fear? Steaphen (talk) 23:59, 12 February 2010 (UTC)


 * You're not satisfied. Noted. Can you provide arguments compatible with policy for making changes to the article? Paradoctor (talk) 00:26, 13 February 2010 (UTC)

I hesitate to use the video for several reasons 1>It is slow moving & 2>rather long & 3>the author states in text & voice that there is no paradox. Using basic arithmetic, we can arrive at a time when Achilles will have passed the tortoise. Using elementary algebra (and given sample speeds and distances), we can derive quite an exact time and position (not with infinite precision, however, because all speeds and distances have limited precision) at which A would catch (and after which would overtake) the tortoise. Using variables for speed and head-start distance is maybe intermediate algebra. We do not really need calculus for any of that. Calculus comes in when we want to find the sum of a diminishing geometric series, and Zeno could be counted as an inspiration for the development of calculus. Calculus (or geometric series math) is necessary for the dichotomy paradox, however. As has been pointed out to Steaphen many times, if you think space and time consist of quanta, there is no infinite series & thus no paradox - but he has some resistance to this that seems to involve infinities between quanta (or something like that) that he advocates on his website. There may even be a WP:COI involved. --JimWae (talk) 06:15, 13 February 2010 (UTC)


 * Okay on the vid, Jim, I understand. As for conflicts and paradoxes and math, all I can say is that if the arrow really can't move, then we are all being tricked by one hell of an illusion!
 * &mdash; Paine ( Ellsworth's   Climax )  21:55, 13 February 2010 (UTC)


 * Sometimes even knowing that we're being deceived won't get rid of the illusion. ;) Paradoctor (talk) 22:28, 13 February 2010 (UTC)

Grünbaum
It is very strange to me that Adolf Grünbaum's classic text Modern Science and Zeno's Paradoxes is nowhere referenced in the article. Sławomir Biały (talk) 12:50, 13 February 2010 (UTC)