Talk:Zeno's paradoxes/Archive 6

Comment by Steaphen
[From Steaphen]: It seems that all editors (excluding myself) at this site lack the ability to ask of  themselves one simple question: "is what I am calculating validated by  experimental data?" -- "Is my method aligned with the fundamentals of the scientific method, of matching or accounting for experimental data  with theory?"

None, based on the above, have followed the basic precept of the scientific method. None of you. Hence my reference to witch-hanging and the like, for none of you have applied sound,  scientific principles in respect of Zeno's Paradoxes.

Despite all the clever calculations, none of you have shown that those calculations  actually apply to tangible physical reality, at least not in the  minutia of physical movement.

Any competent physicist will understand my issue here ... in the minutia of movement physical things  do not follow Newtonian laws of motion ... it is only when sufficient  quantities and distances are covered that macro-Newtonian physics  applies. But in the minutia -- the very point of the paradoxes -- irrespective of the appearance of smooth movement of big things,  Newtonian physics fails. That is the simple, undeniable reality of quantum theory and experiment.

You can calculate until all the witches in eternity are hung or burned, but you won't change the  fundamental fact that Newtonian physics (and the mathematics it is based  on) fails dramatically and conclusively in the minutia of physical  movement.

From this site, one cannot but conclude, based on the above dialogue, that mathematicians go sheepishly and quietly into that  good night.Steaphen (talk)  03:28, 17 February 2010 (UTC)
 * I agree with you that Newtonian mechanics fail to accurately account for experimental reality at  distances that are small enough so that quantum mechanical effects are  not negligible. However, per WP:NOR we can't make a connection in  this article between quantum mechanics and Zeno's paradoxes unless we  are ready attribute it to a reliable source explicitly making that  connection. Gabbe (talk) 12:09, 17  February 2010 (UTC)


 * The reason that people still study Newtonian mechanics, despite knowing they are not perfectly accurate due to  relativity and quantum mechanics, is that Newtonian mechanics are  experimentally verified to be in very close agreement with reality for  objects of reasonable size moving at reasonable speeds.  There is an  enormous amount of experimental data that says that the predictions of  Newtonian mechanics will be extremely accurate for objects the size of a  runner and a tortoise moving at constant slow speeds.


 * So it makes perfect sense to ask whether Zeno's paradoxes cause an  inconsistency in Newtonian mechanics. It turns out that they don't, as  explained in many calculus books. This does not resolve the  philosophical questions behind Zeno's paradoxes, but it does shed light  on them. In the end, that's always the role of formalized models in  physics: to shed light on physical reality by investigating what would  happen in a system that we understand better than we understand physical  reality. &mdash; Carl  (CBM · talk) 12:54, 17 February 2010 (UTC)
 * I can see Steaphen's point in that the map is not the territory. That  is, a model of reality is not reality itself. That we are able to  construct a useful model of reality in which there is no paradox doesn't  mean that there isn't any paradox in the real world. Notwithstanding  this, we can't let the article make a connection between findings in  quantum mechanics and Zeno's paradoxes unless we can find a reliable  source willing to do so, as that would be original  synthesis. Gabbe (talk) 13:27, 17  February 2010 (UTC)
 * Right; but to the extent that we think our model of reality is accurate, the reason that the paradox doesn't hold  in our model can help us see why the paradox doesn't hold in the real  world. This is the role that formal mathematical models (even quantum  physics) have in physics, to help us understand reality by letting us  study a mathematical model instead. In this case, it turns out that the  Newtonian and quantum physics models give different reasons why the  paradoxes don't hold, and both of these help us understand what is going  on in the paradoxes. I agree that the article should include sources if  it mentions quantum physics. &mdash; Carl  (CBM · talk) 13:33, 17 February 2010 (UTC)
 * Precisely. Gabbe (talk) 13:34, 17 February 2010  (UTC)
 * There was an edit conflict, but I'll post a few comments anyway. They were meant as an addition on CBM's earlier  comments.
 * We should keep in mind that Zeno used in his description a classical model of motion, which includes the assumptions that an  object is a point, that in between any two points there is another, and  that motion is on all scales essentially the same.
 * There are roughly two ways to deal with the paradox. The first is to use the  assumptions made in the  model as described by Zeno and show that even  if you start from these assumption the numbers add up. And they add up  mathematically exact, and not just approximately. An alternative, and  complementary approach is to question the assumptions Zeno made. And  modern physics does indeed invalidate some of his assumptions.
 * The first approach is not just historically the most common approach, it is also true to the original argument. If we just point out that his  classical model is dated given our latest understanding of quantum  mechanics we miss the point Zeno tried to make a few thousand years ago.  Ansgarf (talk) 13:59, 17  February 2010 (UTC)

[Fron Steaphen] The witches will still burn, because your filters are simply blinding you to the Reliable Sources who  link Zeno's Paradoxes with quantum mechanics. You can make excuses for being bad scientists (by not rigorously applying the scientific method),  but that doesn't blind others to noting your behaviours, fears and  beliefs. Scientific method has not been applied, or followed. That is the long and short of it (excuse the pun).

Perhaps, given that I have a 'higher' vantage point, I'm able to more easily see the  unsupported, and unsupportable assumptions upon which many here base  their contributions (a bit like seeing that the Earth is not flat, thus  affording one to sail around the globe).

As I have explained previously, and which  is repeated here, when we consider the minutia of movement, things get  'weird' -- and one theory, supported by the quantum evidence, is that  our physical reality simply 'blinks off' very quickly, many times per  second. And that the mathematics (quite valid -- particularly within an expanded framework) merely reveals the superpositions of probabilities  that lie in potential, but unrealised in this probability. Thus the mathematics is still correct, it's just that the mathematical  expressions are not entirely applicable or representative FOR THIS  probability.

If, as good scientists, we take such a theory and bravely ask, "does this explain the facts?" we might be surprised by how well it explains the world we experience. The debate then would shift to a higher-vantage point of considering what else must be going on for  that theory to be valid. Then we would enter a region in which we could more accurately and productively understand,  explain and work the world  we experience.

But alas, it seems there's no real scientists on this site, at least not ones that could be considered bona-fide  scientists who openly seek ideas and theories to fit the facts.

"Ah' but you say, 'that would be original content, or POV or conflict of  interest' ... or similar (ironically, JimWae took issue suggesting a  conflict of interest by me, and this by a mathematics teacher pushing  the validity of mathematics to 'solve' the paradoxes.).

As suggested above, it is exactly your bias that blinds you to the Reliable  Sources who have quite a different, and better take on the nature of  Zeno's Paradoxes.

But of course, you will find arguments why my statement here is wrong, or superfluous or ... whatever. All that you will have demonstrated is your lack of scientific credibility by not  seeking theories to fit the facts.Steaphen (talk) 06:11, 18 February 2010 (UTC)


 * You have a point in criticising the statement "mathematics can be used to calculate  where and when the moving Achilles will overtake the Tortoise". This is  not strictly correct. While it is true in a Newtonian model, in a  quantum physical model we can only calculate the expectation value  of what their position will appear to be when we make an observation.  This is because the value of their position is not a deterministic  variable, but a stochastic one. However, in this case the expectation  value equals the Newtonian value. The wave function of large,  macroscopic objects like tortoises and hominids have very little  variance, so for all practical intents and purposes it will appear  deterministic when we make an observation.


 * To be more accurate we could let the article say "In a Newtonian model we can calculate the exact time when Achilles overtakes the tortoise, but in a quantum  mechanical model we can only calculate the expectation value of what  we will observe this instant to be. This expectation value is equal to  the Newtonian value." But I fail to see why such formality is required  in this article. How is the reader's understanding of Zeno's paradoxes  improved by such rigid strictness? Gabbe (talk) 08:01, 18 February 2010 (UTC)


 * "How is the reader's understanding of Zeno's paradoxes improved by such rigid  strictness?" -- how does the rigid strictness of observing a small  curvature in the Earth improve the belief that the Earth is flat?


 * Within such "rigid strictness" are universes of possibility that are  disallowed by standard Newtonian mechanics. That is, in the 'gaps' of  physicality (and your perception) lay infinite potentials that are not  'realised' in this probability. No small 'improvement' in  understanding.


 * But again, each of you here will disallow alternative views because none of you are sufficiently disciplined or  honest to seek theories that fit the facts.


 * It is enough that you have been alerted to the error in your understanding; you have  stumbled over the facts, and picked up yourselves up as if nothing had  happened. Leave it to others to go noisily and confidently into that  good night, bringing forth new light.


 * Ciao Steaphen (talk) 08:59, 18  February 2010 (UTC)
 * But to "seek theories that fit the facts" is not our task as Wikipedians. Our task is to report what others have  concluded. Wikipedia is not the venue to follow the scientific method.  Alternate views are only admissible in articles if they are
 * attributable (per WP:A)
 * not an instance of undue weight (per WP:UNDUE)
 * In other words, being true is not sufficient for including a statement per Wikipedia's policies. Gabbe  (talk) 10:25, 18 February 2010 (UTC)


 * Hold it, pardner! Verifiability is a central concern of the scientific  method: "Scientific method refers to " ... "  or correcting and  integrating previous knowledge." You could say that the job of  scientists is to produce (scientific) libraries. The job of  encyclopedists is to provide a useful guide to such libraries. What we  don't do is produce substantially new knowledge (there is encyclopedic  research, of course). Paradoctor (talk) 10:43, 18 February 2010 (UTC)


 * (continued at my talk) Paradoctor (talk) 11:29, 18 February 2010 (UTC)

There is more to the scientific method that empiricism. To use logic to check whether an argument or model is logically and mathematically sound is an  essential part of the scientific method. To experimentally check his assumptions and predictions made by models is is another. Even though experiments are thought to be essential to the scientific method, there  is nothing unscientific about the former, in Zeno's case in showing  logically and mathematically that an apparent contradiction isn't a  contradiction. You cannot get more rigorous than that. Just to make it explicit. There might be more to Zeno's paradox, but that doesn't take away that this kind of formal reasoning is not only consistent with the  scientific method, it is an essential part of it. For the tortoise and the runner as described in the paradox you can compute when the latter  passes the former. Ansgarf (talk) 12:58, 18 February 2010 (UTC)
 * It is a verifiable fact that Zeno assumed that the tortoise's and the runner's positions are points (He did not use quantum  superpositions.)
 * It is a verifiable fact that Zeno assumed that inbetween every two points is another point (He did not assume a Planck  length)
 * It is a verifiable fact that Zeno assumed that the entire distance that the runner has to complete is the simple sum of the  smaller distances. Zeno did give no special rules for small distances.  (No discrete quantum leaps)
 * It is a verifiable fact that under these assumptions the sum is finite.  Any textbook will do as reliable  source.
 * BTW: I second Gabbe's bold edit, even thought a previous version of the deleted sentence was  inserted by me. If we take out the particular link between QM and Zeno  from the Proposed Solutions section, since there are no reliable sources  for that particular claim, then it should also be removed from the  introduction. Ansgarf (talk)  13:09, 18 February 2010 (UTC)


 * Dear dear Ansgar,
 * There is no evidence whatsover that Zeno assumed anything. There is only historical  records of what others said of him. The only facts relating to Zeno is  what is currently observable, in existence now, and that is stuff  written on paper or stone or whatever, which we may assume was written  by someone, but we have no evidence of who wrote that material, other  than other bits of paper or stone 'saying' who wrote what.
 * You appear to lack even a modicum of understanding concerning the nature of  'facts'.
 * As for the other, dear me, you lack the simplest of understanding. You may calculate whatever you want, but there is  absolutely no evidence (or tight correlations) between what you are  calculating and what you assume those calculations relate to, or are  correlated with. Could be angels on pinheads, for all the evidence I've  seen.
 * From the contributions that have been provided by others, all of you would be accepted into the flat-earth society, or the brethren in  Galileo's time, because you each have opinions and superstitions that  are unfounded or unsupported by, or incongruent with the facts.
 * Is it not galling for you to realise that readers of this page will have  solid grounds for recognising that you're no better than the priests in  Galileo's time, or flat-earth believers -- due to your blind adherence  to superstitions that cannot be verified in, or strongly correlated with  fact?
 * Steaphen (talk) 00:16, 21 February 2010 (UTC)


 * But Newtonian mechanics has been thoroughly verified by centuries of experimental evidence. For  object the size of a tortoise or runner moving at the speeds typical of  these things, the experimental evidence is that the predictions of the  Newtonian model will be extremely accurate. It's hardly a "superstition"  that the objects we observe in everyday life act in accordance with  Newton's laws. &mdash; Carl  (CBM · talk) 00:29, 21 February 2010 (UTC)


 * Dear Carl, Ansgar, Paradoctor and the Brethren (of mathematicians, and  wannabe mathematicians)
 * The approximate value and efficacy of Newtonian mechanics has never been questioned or disputed.
 * Zeno's Paradoxes involve explaining movement in detail, in the minutia, which  involves quantum mechanics, for all objects, irrespective of size. To  say otherwise is just sloppy thinking, and that's being polite.
 * This page will, or should prove to be a classic textbook example of sloppy  thinkers believing opinions and superstitions as 'fact'. The wonderful  irony is that the contributors are supposedly scientists. That will be  the icing on the cake for students in centuries to come. Delicious  ironies that beggar belief.
 * "And to think", they will say, "that they had their fingers on the button. How on earth did civilisation  survive." (that's the optimistic probability that may not eventuate, due  to the ignorance and superstitions of this age.)
 * To highlight the irony: The extent to which the Earth is flat is the extent to which  Newtonians mechanics (and the mathematics thereof)solves Zeno's  Paradoxes. Both are crude limited-dimension perceptions of a deeper  multi-dimensional reality (just like a motion picture film provides a  compelling and believable 2D illusion of a 3D reality).
 * The illusion and crude efficacy of Newtonian mechanics has never been  disputed.
 * You've continually been distracted by the action on the screen, not asking questions that would lead you to the projector, then  to the director, the production process, and marketing thereof.
 * Steaphen (talk) 01:05, 21 February 2010 (UTC)


 * "Zeno's Paradoxes involve" ... "quantum mechanics": WP:PROVEIT  Paradoctor (talk) 02:07, 21  February 2010 (UTC)
 * [in reply to Paradoctor, for Ansgarf's benefit!]-- It is, once again, highly disingenuous of you to ask that I  prove something that is not in contention. What is in contention is the  use of Newtonian mechanics in solving Zeno's Paradoxes. The onus is upon  you, and the rest of the brethren to prove correlations (at and below  the Planck length) when discussing movement of physical things. Do any  of you have even an ounce of scientific integrity? If those correlations  of theory with fact involve quantum mechanics (which I believe does)  then so be it. But start first with integrity, honesty and discipline,  then see where that leads you.Steaphen (talk) 02:13, 21 February 2010 (UTC)
 * "in contention is the use of Newtonian  mechanics in solving Zeno's  Paradoxes": If there are claims in the article not sourced to reliable  sources, or not actually supported by their sources, kindly point them  out, so your concern can be addressed. Paradoctor  (talk) 02:59, 21 February 2010 (UTC)


 * [in reply to Paradoctor, for Ansgarf's benefit! -- from the main article,  as at Feb. 21, 3.00pm Australian EST]"These works have resolved the  mathematics involving infinite processes, including Zeno's, and the  paradoxes no longer present any mathematical problems.[14]" -- Uhm,  irrespective of whatever Reliable Sources say regarding the burning of  witches, where is a Reliable Source who says we can apply mathematics  all the way down, below the Planck length and shorter? Once again, it is  sloppy thinking (Pseudoscience) to make such statements. The  assumption (unsupported = superstition) is that the mathematics that you  calculate actually relates to something physical. I have seen no  experimental data confirming that it does. But don't worry, the brethren  will be proud of you.Steaphen (talk) 03:58, 21 February 2010 (UTC)
 * Dear Steaphen,
 * I never mentioned Newtonian physics in this tread.
 * Sloppy thinking is to confuse Newtonian mechanics with calculus and geometric  series.  All the reliable scientific publication on quantum mechanics  used mathematics, even infinite series mathematics like calculus or  integrals. This includes also every single paper you quoted yourself in  this and previous discussions. Either you didn't read them, or you do  not know how to recognise integrals and differential equations over real  or complex numbers. Neither speaks for your scientific credentials.
 * I contend that Zeno, who lived 4 centuries BC, did not refer to quantum  mechanics in any meaningful way. Provide a reliable source to the  opposite, or I'll have to assume that you just made that link up.  If  the link wasn't contended before, now it is.
 * I made some fairly explicit statements, about what Zeno paradoxes assumes. And you  are right to ask for sources. The source is  . It does not mention  quantum mechanics. It does nowhere mention that motion over small  distances is different from motion at large distances, and it nowhere  says you cannot add small numbers the same way you add other numbers.  Please provide an alternative source to show that Zeno did talk about  any of these. Ansgarf (talk)  04:54, 21 February 2010 (UTC)
 * Steaphen said "The assumption (unsupported = superstition) is that the mathematics that you calculate  actually relates to something physical. I have seen no experimental data  confirming that it does.". If you haven't seen that the mathematical  results relate to something physical, you might want to check Ehrenfest  theorem. It give a very clear relation between what is computed and what  is observed. One that has been experimentally verified.  Ansgarf (talk) 08:39, 21 February  2010 (UTC)
 * Dear dear Ansgar,
 * You know, the silliness keeps on getting sillier. in regards to Ehrenfst's theorem, it deals with (in  part) the "expectation value" of quantum mechanics --> "Quantum  physics shows an inherent statistical behaviour: The measured outcome of  an experiment will generally not be the same if the experiment is  repeated several times."
 * So, in the minutia of movement, you can 'expect' your nose to be where it is, in order for you to pick it. But  you can't precisely calculate it, otherwise you wouldn't need to expect  it, would you.
 * Alice in wonderful stuff. As for your other comments, Ciao Steaphen (talk)  09:25, 21 February 2010 (UTC)
 * I take your first statement to mean that you have nothing but facetious remarks to back up your claim that  infinite series mathematics implies Newtonian mathematics. Fair enough, I  assume it was just a slip of the tongue, and you simply didn't mean the  claim.
 * With respect to Ehrenfest, you are right with the observation that in Quantum mechanics everything are distributions. Your  nose is a distributions as well, and not at a single point. Don't  forget that. And mathematics deal with distributions perfectly fine.  Ehrenfest's theorem says that the centre of your nose behaves exactly  like a classic particle. And that has been experimentally confirmed,  admittedly not for your nose, but for plenty of other particles.
 * Back to the topic, can still provide a source that states that Zeno was  talking about quantum mechanics, or even better a source that confirms  that according to quantum mechanics the geometric series does not  converge. Ansgarf (talk) 10:53,  21 February 2010 (UTC)
 * [in reply to Ansgarf, for Ansgarf's benefit!] Uhm, pray tell, you can do Newtonian mechanics without  calculus or geometric series?
 * You have evidence that you know ALL reliable scientific publications on quantum mechanics.
 * As for the remainder of your comments -- you're teasing me again, aren't you,  Ansgar. There's that question again that I need to keep asking myself  ... "he's not serious, is he?". Steaphen (talk) 05:54, 21 February 2010 (UTC)
 * So the fact that Newtownian Mechanic uses calculus and infinite series proves what?  Quantum mechanics uses calculus and infinite series. The Schroedinger  equation is partial differential equation, for example.
 * But thanks for your question. It shows that you are either unfamiliar, or forgot  for a moment what equality entails. When someone says "Do not confuse A  with B" it is meant to say "A is different from B". To show show that  they are actually the same, you do not only have to show that A impies  B, but also that B implies A. So, you would have to show that infinite  series implies Newtonian mechanics. Which might be difficult for the  simple reason that comparing both is a category mistake to begin with.  But if you can show that infinite series mathematics implies Newtownian  mechanics, please do so.
 * I haven't read all publications on quantum mechanics. You got me. So to prove the opposite, just show me  one reliable publication on quantum mechanics that does not rely on  infinite series mathematics. Just one. You have read a scientific  publication on quantum mechanics, haven't you?
 * I am serious, and I took your comment about my claims on what Zeno assumed fairly serious  too. It seemed that you were implying that I am wrong about what Zeno  assumed. Can you please provide a single source that confirms that I am  wrong about it. Just one source confirming that Zeno paradoxes refer to  concepts from quantum mechanics. Or that Zeno paradoxes do not assume  that in-between any two points there is another.  Ansgarf (talk) 08:39, 21 February  2010 (UTC)


 * I believe Steaphen disowned the statement that explaining Zeno's paradoxes  involves quantum mechanics. Paradoctor (talk) 13:02, 21 February 2010 (UTC)

ArBreak 1
May I first remind everyone that this Talk page is for discussing specific changes to the article. It's neither a discussion forum on the general topic of Zeno's paradoxes, nor a venue to talk  about the qualities of our opponents. See WP:NOTAFORUM, WP:TALK and WP:NPA.

Now, Steaphen, could you be more explicit in what your objection is with the statement "These works have resolved the  mathematics involving infinite processes, including Zeno's, and the  paradoxes no longer present any mathematical problems." When reading it, I don't interpret this as saying that by calculus we can resolve every  aspect of the paradoxes. Together with the rest of the article, I interpret it as saying that calculus can solve the mathematical  conundrum of whether the sum of an infinite number of terms can be  finite, a conundrum which is inspired by (and often conflated with)  Zeno's paradoxes. Do you have a suggestion on how we can improve the wording of the article? Gabbe (talk) 10:12, 21 February 2010 (UTC)


 * I adressed the concern Steaphen  raised above. Rudin's book doesn't seem to even mention either Zeno or  paradoxes, and Rudin himself appears not to have published in the  field. The reference was inserted with  this edit in response to a citation needed tag. If nobody comes  up with sources, I suggest removing the paragraph in a few weeks.  Paradoctor (talk) 12:46, 21  February 2010 (UTC)


 * It is, again, quite simple. Any statement that states or infers that mathematics can be used to plot, explain or  predict the movement of physical things, in the minutia of movement, is,  according to the evidence at hand, as at Feb. 22, 2010, an assumption,  without any evidence to support that assumption. While empirical  evidence supports the use of calculus for large bodies within certain  limits, the efficacy of that use disappears when we look in detail at  the process of movement (the detail of which is entirely relevant to the  issue of Zeno's Paradoxes). Just as we may say "the Earth is flat"  within certain limits, so may we use mathematics to plot and predict the  behaviours of large bodies -- within certain limits. Is anyone here  arguing that the Earth is flat? Why then do you persist with the  nonsense of arguing that you can apply crude approximations to the  minutia of physical movement.
 * As stated above, to do so is, I'm sorry to say, just plain stupid. It is the height of idiocy to assert  that one can use mathematics to deterministically (precisely) plot and  predict the minutia of movement of physical objects (runners, hares,  arrows etc) when empirical data (and the quantum theory) conclusively  and repeatedly shows that you cannot.Steaphen (talk) 00:56, 22 February 2010 (UTC)
 * The question is not whether QM is deterministic or not - it is not - but that doesn't  mean that you can not use math to compute the evolution of a  distribution with mathematical precision. And it happens that the centre  of the distributions behave deterministically like classic particles.  But, before we get into details, do you have reliable sources for your  claims. I do have sources for mine. And do you have sources that links  this to Zeno. I am not quite sure if you disowned your earlier claim  that Zenos paradoxes are about QM. If you did please confirm this.  Ansgarf (talk) 01:30, 22 February  2010 (UTC)


 * (Steaphen) "Any statement that states or infers that mathematics can be used": Is that leading to any suggestion for a  specific change to the article? If not, this would constitute  WP:SOAP.
 * (Steaphen) "just plain stupid. It is the height of idiocy": Is this WP:CIVIL?
 * (Ansgarf) "before we get into details", please explain which (proposed or actual) changes to the  article you're discussing.
 * (Ansgarf) "have sources for mine": Where are they?
 * (Ansgarf) "you disowned your earlier claim" ... "please confirm this": Yes, please. Paradoctor  (talk) 01:51, 22 February 2010 (UTC)


 * With respect to the requests made to me. First I am discussing Steaphen's  request to remove any statement that mathematics can be used. I am  arguing that mathematics can be used, and that QM is no reason not to  use mathematics, and not even a reason not to use a classic description,  even if we were discussing the movement of actual objects, rather than  those of hypothetical tortoises.
 * Sources that confirm that the centre of the wave function behaves like a classical particle are    and .  Note, that the second article says in the very last line "...the centre  of a wave-packet always moves like a classical particle". These are  lecture notes that I also quoted in the last   mediation  attempt.
 * I assume that the other comments are addressed at Steaphen and for him to respond to. Ansgarf (talk) 02:56, 22 February 2010 (UTC)


 * I clarified which remark was addressed to whom, sorry for the confusion.
 * The Takada reference says "In this sense, we can say that quantum mechanics involves the classical mechanics.". That is not specific  enough IMO. The Fitzpatrick source is good, I'll add it. (Please let's  bury the mediation and arbcom requests, ok?)
 * "I am arguing": Please don't, that didn't get anywhere over the past months.  —Preceding unsigned comment added by Paradoctor (talk • contribs) 13:19, 22 February 2010

Again! While empirical evidence supports the use of calculus for large bodies within  certain limits, the efficacy of that use disappears when we look in  detail at the process of movement (the detail of which is entirely  relevant to the issue of Zeno's Paradoxes).

In the minutia of movement, particles (the stuff that makes up arrows, hares and people  etc.) do not move deterministically. If you can show otherwise, -- that you can use any mathematical method to precisely plot and predict the  movement of things when they move in small increments -- then  congratulations, you have yourself a Nobel Prize in physics.

The use of calculus or any mathematics that infer or state that things will  be precisely at a particular location at a particular precise time is  demonstrably WRONG. To say otherwise in the face of direct evidence is -- and again I must apologise, but truth be said -- to deny  reality is  just plain stupid. What part of the experimental data do you have issue with? What part of repeatable, independently verifiable evidence is causing you trouble? What part of reality is causing you lot to stick your heads in the sand? What are you frightened of?

And Ansgar, you genuinely honestly can't be serious; "Sources that confirm that the  centre of the wave function behaves like a classical particle are" .. "LIKE a classical particle?" LIKE? Infinite-series solutions require absolute, unremitting "AS' not like, not 'close enough' or  'approximately' or anything other than "AS" perfectly, completely and  without error.

As we say in Australia, fair dinkum, you wouldn't believe it if you didn't actually read it here. Incredible. Steaphen (talk) 07:47, 22 February 2010 (UTC)

Re "First I am discussing Steaphen's request to remove any statement that mathematics can be used." Request? for what? Again, for those who are having difficulty in understanding: While empirical evidence supports the use of calculus for large bodies within  certain limits, the efficacy of that use disappears when we look in  detail at the process of movement (the detail of which is entirely  relevant to the issue of Zeno's Paradoxes).

For those who are again slow in understanding, the mediation was called because of those  issues as stated in the mediation, the bulk of which have not been  addressed.Steaphen (talk)  07:57, 22 February 2010 (UTC)


 * So on and on the discussion goes, circling around forever and ever.  Common sense must rule in the end.   So what is the common sense of this discussion?  Where does common sense  lead this long, drawn-out discussion?  Common sense leads us to accept  that the arrow leaves the bow and hits the target.  Common sense leads  us to accept that Achilles catches and passes the tortoise.  Common  sense leads us to accept that quantum mechanics is a part of reality, or  else we couldn't be here typing away on these awesome computers.  Where  does this leave us?  It is natural for a mathematical mind to deal with  all of these common-sense acceptances.  The mathematical mind will  figure, figure and figure some more.  Common sense leads the mathmatical  mind to analyze the movements of the arrow, the racers, etc. with as  much precision as possible.  Common sense tells the mathematical mind  that such an analysis is valid, true and a part of reality.  The sad  thing is that common sense leads the non-mathematical mind to reject  these analyses with every tool possible, including the uncertainty  principle.  Apparently, common sense must therefore lead us back to  basics.  If there is a Wikipedia policy or guideline that prohibits the  inclusion of mathematics in this article, then common sense must lead us  to remove every single reference to mathematics in this article.  And  that's the bottom line.  Produce such a Wikipedia policy or guideline.   If one cannot produce a Wikipedia policy or guideline that prohibits the  use of mathematics in this article, then common sense must lead one to  accept the inclusion of said math.  Simplistic?  Absolutely.  Common  sense usually is.
 * &mdash; Paine  (Ellsworth's  Climax )  09:08, 22  February 2010 (UTC)
 * Paine, sorry to go around one more time in a circle, but I cannot help to respond to some of the points that Steaphen  raised.
 * Steaphen, you said earlier that in quantum mechanics everything is a distribution, and I agree. The wave associated with  every particle is a distribution too, and distributions can evolve  deterministically, even if the distribution itself is not.  This extends  to Ehrenfests theorem too. The exact mean of the wave behaves like a  classic particle. Without going into the semantics of the word "like",  but it is used here with the meaning "in the manner of".
 * I am honoured that you think this would deserve the Nobel price, but I am  afraid that we are almost a century late. Every element in Heisenberg's  matrix mechanics, an equivalent model to the Schroedinger equation,  satisfies the classic equations, and he got a Nobel price for that.
 * Unfortunately you forgot to address my question. Do you still claim that Zeno's  paradoxes refer to Quantum mechanics in a meaningful way?  And do you  have sources for that?
 * It seems that your latest request dismisses what I said about you alleged request. To clarify. Do you want that the  statement to be removed that mathematics can be used to compute when  the runner can pass the tortoise?   Ansgarf (talk) 11:11, 22 February 2010 (UTC)

Bold edits
in an attempt to allay everyone's concerns. I added two sections clarifying how motion is explained in the Newtonian and quantum models. I know that the two sections are completely uncited, and I have all  intention of adding references. What I wonder is whether editors here think this is sufficient or completely inadequate. Gabbe (talk) 09:15, 22 February 2010 (UTC)


 * Very happy with it, a big step in the right direction. We'll need sections on  Aristotelian physics, pre-Aristotelian physics (Zeno was born a  century earlier), and several other approaches, such as digital  physics. Also, there will have been attempts at reconstructing the  philosophy of space and time Zeno was criticizing, though this probably  belongs with pre-Aristotelian physics. Also note that quantum physics is  an umbrella term. There are deterministic  interpretations of quantum mechanics. OTOH, I recall reading about  accounts in which not even the concept of single particle is  meaningful. Paradoctor (talk) 11:15, 22 February 2010 (UTC)


 * There was another of your bold edits that concerns me.  I added a reference  with  this edit, and you removed the claim and citation with  this edit.  That was a valid claim and well-referenced.  So what, may I  ask, moved you to so boldly remove it?
 * &mdash;  Paine (Ellsworth's  Climax )  09:26, 22  February 2010 (UTC)


 * I removed it because
 * (a) it was a high school science textbook. While not unreliable per se, it  doesn't qualify as a "high quality" source on the topic. An academic  book on philosophy or article in a peer-reviewed journal would have been  much better.
 * (b) it mentioned the resolvability of Zeno's paradoxes in passing. Calculus was the topic of the book, not Zeno's  paradox.
 * (c) it was talking about a mathematical model. It didn't discuss, explicitly, whether this mathematical model had real world  applicability, which was the actual point of contention.
 * Therefore I removed it. Gabbe (talk) 09:44, 22  February 2010 (UTC)


 * Regarding your three points:
 * (a) a high school science book does qualify as a "high quality" source  in my opinion, because I would hope that our children are being fed  facts and not fantasies or fiction.
 * (b) aside from the fact that there are many reference sources on WP that mention an  article's topic "en passant", this text did more than just give a  passing treatment of Zeno's paradoxes.  It did indeed show how to solve  the Achilles/tortoise motion paradox mathematically, it showed the  "higher math" involved in coming to this solution.
 * (c) the only claim referenced was "While mathematics can be used to calculate where  and when the moving Achilles will overtake the Tortoise", and this was  the only "point of contention".  The cited reference did reliably  confirm that math can be used to show that Achilles would indeed catch  and pass the tortoise.
 * Therefore in accordance with WP:PRESERVE and WP:BURDEN, I shall replace it in the article.   Please discuss further before removing it.
 * &mdash;  Paine (Ellsworth's  Climax )  09:15, 23  February 2010 (UTC)
 * You could just as easily have cited "String  Theory for Dummies", which says the opposite. It asserts that while a  geometric series is traditionally brought forth as a solution, it does  not resolve the physical aspects of paradoxes. My point is not that we  should use "[...] for Dummies" books as sources. My point is that there  are sources whose quality are higher than both "String Theory for  Dummies" and your textbook, and those are the sources we should be  relying on in this article. Gabbe (talk) 11:19, 23 February 2010 (UTC)


 * (ec) The source does not say that the Achilles/tortoise paradox can be  "resolved", it merely shows how mathematics can be used to show when and  where Achilles will pass the tortoise.  As for higher or lower level of  quality, I maintain the high quality of this reference for the reason I  already stated.  I do not contend that there may or may not be higher  sources, I simply state that inasmuch as the claim being referenced,  this source is "high-quality enough".  It is a text used to educate our  young, it is a text that shows in detail how the claim is a part of  reality, therefore it is of high enough calibre to fall within the  policies and guidelines of Wikipedia.
 * &mdash;  Paine (Ellsworth's  Climax )  11:22, 23  February 2010 (UTC)


 * Textbooks not specifically on Zeno or paradoxes are useless for our purposes. WP:RS is always a question  of context. Paradoctor (talk) 11:48, 22 February 2010 (UTC)

Of Pandas and People is also a "text used to educate our young", but  that doesn't make it a reliable source. Furthermore, nobody has disputed that in the Newtonian model of reality (that is, where spatiality and  time are both infinitely divisible and particles deterministically  occupy specific points in space at specific points in time) "mathematics  can be used to calculate where and when the moving Achilles will  overtake the Tortoise". The textbook provides ample proof for that. The point that that Steaphen was disputing was whether this is also true in  the real world, and not just in the Newtonian model of the world. And on this question the current source is silent. Gabbe (talk) 12:06, 23 February 2010 (UTC)


 * The editor who suggested the source in a conversation above thought it was reliable, I think it's  reliable, you don't think it's reliable.  Rather than go 'round and  'round about it, feel free to test the source's reliability with WP  administration.  Our conversation (yours and mine) is not about models  of reality, it is about whether a claim you removed along with its  reliable source does or does not improve this article.  I put it back in  because I believe it improves this article.  I gave my reasons for  doing so.  You have not convinced me that I was wrong to do so.  You  appear to be trying to get me to discuss the philosophical questions,  while I am trying to discuss the mechanics of whether or not a claim can  be substantiated by a reliable source.  The source is reliable, the  claim is therefore reliably sourced, and unless the WP administration  rules against the source's reliability, then there is no reason to  remove it nor the claim.
 * &mdash; Paine  (Ellsworth's  Climax )  12:29, 23  February 2010 (UTC)


 * Allow me to that while I don't think the source  is unreliable per se, it is not a "high quality" source.  WP:SOURCES says that "[t]he most reliable sources are usually  peer-reviewed journals; books published by university presses;  university-level textbooks; magazines, journals, and books published by  respected publishing houses; and mainstream newspapers." The textbook  doesn't qualify as one of those and I seriously doubt that a discussion  at WP:RSN or a similar forum would conclude otherwise.
 * Furthermore, neither I nor anyone else here has disputed that mathematics can solve  the mathematical problem of when Achilles overtakes the tortoise.  What Steaphen disputed was whether calculus addressees the  non-mathematical aspects of the paradox, and as such the statement  (even though supported by a source) is vague and ambiguous. What he said  was that the article as it now stands misleadingly implies that it is  possible to calculate this in the real world, and not just in  mathematics, and in that case the source doesn't "directly support  the information as it is presented" (per WP:V and WP:NOR).  Gabbe (talk) 16:14, 23 February 2010  (UTC)


 * (Paine) "I believe it improves this article. I gave my reasons for doing so": I'm afraid I don't see where. The argument above  turns mostly about the sources. I agree that the statement is  sufficiently supported, even though there will be more authoritative  sources. The statement is about the admissibility of using mathematical  arguments. This is something that we need to source to the literature  about the paradox, i. e. we need sourced that discuss this question.  I've seen comparable discussions in the literature about the twin  paradox, where some argue that the ability to predict the ages of the  twin solves the paradox, whereas others maintain that this does not  address the question of the validity of this approach. Sources, that's  what we need. Paradoctor (talk) 17:58, 23 February 2010 (UTC)


 * You apparently attached the second sentence to the secondary clause in the  first sentence.  The entire two sentences read, "I put it back in  because I believe it improves this article.  I gave my reasons for doing  so.  (I.e., I gave my reasons why I put it back in.)  My reasons  for believing that the claim improves the article have to do with the  way the thought "leads in" to the further thoughts about philosophy and  how mathematics does not actually dig deep enough to resolve Zeno's  paradoxes.  Editor Jim Wae in an above conversation came up with many  sources, all good.  I chose the textbook because I wanted to show that  the math involved, while a higher math, is yet a very basic math.  There  are plenty of sources.  Pick one.
 * &mdash;  Paine (Ellsworth's  Climax )  00:30, 24  February 2010 (UTC)


 * "The most reliable sources are usually" etc. etc.  And I am very certain that the Wikipedia  administration, who test each source on an individual-merit basis, would  allow this high-school text in the context of this article and this  claim.  So as I said, feel free to test the source with them.  And  please don't say what you didn't dispute.  You most certainly did show  what you disputed when you boldly removed the claim AND the reliable  source.  As for Steaphen separating mathematics from the real world, ask  him how he would be typing on his real-world computer if it weren't for  the fantasy math of quantum mechanics?  There isn't always a  difference between math and the real world EXCEPT in the minds of some  sadly misinformed non-mathematicians!  And there is no difference  between math and the real world in this case.  Because in the real  world, Achilles actually does catch and pass the tortoise!
 * &mdash;  Paine (Ellsworth's  Climax )  00:30, 24  February 2010 (UTC)
 * I'm not sure what you mean by "the Wikipedia administration", so I don't know whom you want me to "test  with". I've never said the I disputed the veracity of the statement.  What I'm trying to say is that whether mathematics can be used to  calculate when Achilles overtakes the tortoise has been debated on this  article talk page for about five years, and in the philosophical realm  for about two millenia. It's a bit like inserting the statement  "abortion is wrong" to the Abortion article. Putting it like that is  bound to be highly contentious (even if sourced to a high school  textbook), and needs to be either properly clarified or substantiated by  a very reliable source. One way of clarifying would be by saying "the  Pope says that abortion is wrong" or "a survey shows 40% of Americans  think abortion is wrong" or similar. And again, nobody here (that I know  of) has disputed that Achilles actually manages to pass the tortoise,  what was disputed was whether mathematics can calculate when this  happens in the real world. Gabbe (talk) 13:32, 24 February 2010 (UTC)


 * Your link above to Reliable sources/Noticeboard is precisely what I  meant by WP administration.  If they say the source is not good enough,  then I would, of course, abide by that ruling.  Up to you as to whether  or not it should be submitted and tested.  As for the lustrum-long  debate here and the millennia-long global debate, we might want to  consider that these debates might never end.  What should end, in my  opinion, is the debate about the claim as a valid claim, supported by  reliable sources and an improvement for this article.  The fact that the  claim is and has been debated, the fact that the claim may or may not  be true and a part of reality, these ideas simply give the article and  the paradoxes a more "magnetic personality".  And I think it lends an  attractive polarity to the philosophical ideas that follow the math  claim.
 * &mdash; Paine  (Ellsworth's  Climax )  22:45, 25  February 2010 (UTC)


 * How about including Peter Lynds paper as source . I would  however keep in addition the textbook source, and add another textbook  .  In the case of Zeno's paradoxes you will be hard pressed to find a good  treatment outside of a textbook. Scientific papers usually mention this  in passing, since it is assumed to be high school knowledge. So, if we  include a more reliable source gives a cursory treatment, the textbook  sources allow an interested reader to get also the  details.Ansgarf (talk) 00:30, 24  February 2010 (UTC)


 * Thank you, Ansgarf!
 * &mdash;  Paine (<font size="2"  color="darkblue"  face="Ariel">Ellsworth's  <font  color="blue" face="Ariel">Climax )  06:45, 24  February 2010 (UTC)
 * I'm not sure quoting Peter Lynds is a good idea, he doesn't represent the mainstream and therefore I think  mentioning him would violate WP:DUE. The Lee paper, however, is a  very reliable source. Gabbe (talk)  13:32, 24 February 2010 (UTC)
 * He published in Foundations of Physics. Not mentioning him would violate WP:DUE, IMHO.  Also, the abridged version cited in the article has  two journal citations, I think you'll be able to find some for the  main entrée, too. Paradoctor (talk) 21:02, 24 February 2010 (UTC)

With respect to the latest bold edit on Newtonian and Quantum mechanics. The content is good, but I think it that including it gives undue weight to  the assertion that Zeno's paradoxes are about the Newtonian versus  Quantum mechanics. If this could be condensed to one or two sentences I would not object including it, but even then there is the risk that it  could be considered original research. Ansgarf (talk) 11:17, 22 February 2010 (UTC)


 * This is addressed by my latest edits, I trust it? ;)  Paradoctor (talk) 11:42, 22  February 2010 (UTC)

Front and center, please!
This is addressed to Ansgarf and Steaphen. This talkpage is drowning in endless, fruitless discussion. If this doesn't stop, I'm going to hurt myself. It is obvious that you two don't know how to speak with one another. I'm not assigning blame, I'm looking for a solution. The screen below is blank, use it. (The order of the sections is alphabetical, nothing else) Paradoctor (talk)  13:48, 22 February 2010 (UTC)
 * 1) I  suggest that you state here which specific changes you want to see,  as in "add, remove, change" the phrase "...".
 * 2) Do not argue for  your requests, provide sources supporting these changes, and provide  references to policy, where appropriate.
 * 3) I furthermore suggest that  you do not reply to each other in this section, but only to  comments by others. If nobody else does, I will provide comments for all  requests so you have someone you can respond to.

Specific changes proposed by Ansgarf
Unrelated to Steaphen assumed proposal, I would propose a few changes.


 * The paragraph "Another proposed solution is to ...". I previously gave  reasons why this statement is accurate, but unless there are sources  that confirm this link it is probably original research.  I wouldn't object to removing this paragraph, unless  we find a reliable source that links this to Zeno's paradoxes.
 * Have cned the paragraph. Paradoctor (talk) 00:32, 23 February 2010 (UTC)


 * I propose to delete the heading "Physics", if the remainder of the section is  reduced in size as proposed below.
 * as long as this part of the article is short, we can do without it. If nobody objects, I'll remove it later on.  Paradoctor (talk) 00:32, 23  February 2010 (UTC)


 * Paragraph "In the 18th and 19th centuries, the Newtonian model".  I would keep the paragraph as is,  and there should be even textbook sources to support it.
 * I concur. Paradoctor (talk)  00:32, 23 February 2010 (UTC)


 * Paragraph "By the early 20th century, the Newtonian model ...".  I propose to remove the entire paragraph. The paradoxes are  not about Newtonian vs. Quantum mechanics.
 * I oppose removal. Sources are missing, that is the real problem.  Paradoctor (talk) 00:32, 23  February 2010 (UTC)
 * The current paragraph has no direct  and sourced link with Zeno's paradox. Ansgarf (talk) 03:41, 23 February 2010 (UTC)
 * Thinked again. The entire paragraph is unsourced. If Gabbe does not object, I  think we should kill the entire Physics subsection, and reinsert only  sourced statements. Paradoctor (talk) 05:02, 23 February 2010 (UTC)
 * I don't feel any attachment to it, so feel free to delete. Remember, I did  explicitly point to WP:BRD while inserting it. However, if the  only objection was that it was unsourced I can have a sourced  equivalent reinserted by the end of the week. Gabbe  (talk) 07:08, 23 February 2010 (UTC)
 * I love sourced edits. ;) Paradoctor (talk) 20:34, 25 February 2010 (UTC)

''
 * Paragraph "Some philosophers[15][16] say that the mathematics  ...". The central  part of this paragraph is fuzzy and rehashes earlier statements. I  propose to make it a lot shorter. My proposal is to change it to:
 * Some philosophers say that the mathematics does not address the  central point in Zeno's argument, and that solving the mathematical  issues does not solve every issue the paradoxes present; in particular  how to finish'' an infinite number of tasks. Philosophers   say that calculus does not address that question, and hence a solution  to Zeno's paradoxes must be found elsewhere.

''
 * Update: I found a source saying that "Some asserted that space and time are  not, in fact, infinitely divisible, and moved on"  .  Maybe we can keep the paragraph "Another proposed solution is to ...",  but I would move it to the end of the section.  —Preceding unsigned  comment added by Ansgarf (talk •  contribs) 23:39, 22 February 2010  (UTC)
 * I don't know whether Brown can be called a philosopher.
 * The source you cite is a textbook, has no references, it is not clear who on the advisory board has overseen that  page, and while I didn't check their work, none of them looks like an  expert on paradoxes or the philosophy of space and time. I think we can  do much better than that.
 * As said, if we cannot find a reliable source, I am not opposed to removing the  paragraph.Ansgarf (talk) 03:41,  23 February 2010 (UTC)
 * "fuzzy": Yes, but the text you want to delete makes sense for me, I think it should be cned.  Paradoctor (talk) 00:32, 23  February 2010 (UTC)
 * The sentences make sense to me too, but they are rehashing the paradox, and introduce a problem with sourcing, by  introducing a statement that looks like an original contribution.  My  proposal was the shortest possible way to get rid of the unsourced  tag. How about the following
 * Some philosophers say that the mathematics does not address every  issue the paradoxes present. Rather than that the sum of an infinite  number of terms must itself be infinite, the central point in the  paradoxes was how to finish'' an infinite number of tasks.  Philosophers   say that calculus does not address that question, and hence a solution  to Zeno's paradoxes must be found elsewhere.
 * Ansgarf (talk) 03:46, 23 February 2010 (UTC)
 * I would prefer to tag, but I will not oppose the suggested change.  Paradoctor (talk) 20:34, 25  February 2010 (UTC)

Specific changes proposed by Paradoctor

 * Discussing actual changes with Ansgarf made me realize something I should have seen much sooner. This topic is a battleground.  It has been so for about 2400 years. In light of this "revelation" it  makes no sense to accept any unsourced statements into article,  they're going to be challenged anyway. I'll review the article and make a  proposal on how to best inspissate the existing material.  Paradoctor (talk) 05:08, 23  February 2010 (UTC)

Specific changes proposed by Steaphen
(initial message moved here from section below Paradoctor (talk) 13:35, 25  February 2010 (UTC))

Below are listed some of the issues that are revealed if one applies  good scientific/journalistic standards:

1. "Zeno's  paradoxes were a major problem for ancient and medieval  philosophers"  This infers it is no longer a problem. What evidence or Reliable Source  says it is no longer a problem? And if some wish to argue there is no  inference, then why state they were a problem, if  they are still a  problem now? That would be equivalent to saying 'I was married', when  remarking on one's present marriage =>  nonsense
 * Not my understanding, but I'm not a native speaker. When you say "infer", you probably mean "imply"? Can  you reformulate the sentence to avoid the connotation you see?  Paradoctor (talk) 20:28, 25  February 2010 (UTC)

2. "while many philosophers still hesitate" ? says who. Hesitate? I don't think so. What Reliable Source says  'philosophers still hesitate', or what survey of professional   physicists, who are 'at the coalface' at studying this problem, can you   cite that confirms they 'hesitate' in regards to this matter?
 * Tagged the statement cn rather than who?, statements about "many" are terribly hard to source from primary sources. The latter is  appropriate for statements involving "some", "there are those", ...  Paradoctor (talk) 20:28, 25  February 2010 (UTC)

3. "Modern calculus achieves the same  result, using more rigorous methods (see convergent series, where the  "reciprocals of powers of 2" series,  equivalent to the Dichotomy  Paradox" -- what Reliable Sources confirm  that 'reciprocals of ..." are  equivalent in actual reality in regards to  the Dichotomy Paradox?
 * Tagged the statement cn, but I have no problem if the parenthetical remark gets deleted.

4. re quantum mechanics being a  'A competing theory" to Newtonian mechanics ... since when is it a   competing theory? Quantum mechanics simply eclipses Newtonian  mechanics,  in a similar manner that 'round earth' theories eclipse  'flat-eath'  approximations.
 * Competing in the sense that both have been used as fundamental theories of nature. What change in wording do  you suggest? Paradoctor (talk) 20:28, 25 February 2010 (UTC)

5. yet again, "While mathematics can be used to calculate where and when the  moving Achilles will overtake the Tortoise," - what Reliable Source  actually confirms this, beyond  mathematicians fiddling with equations  and assuming this actually  relates to the minutia of physical movement?
 * Tagged citations as request quotation. Paradoctor (talk) 20:28, 25  February 2010 (UTC)
 * I do not understand why, as indicated by the tag, the claim  needs quotation on talk to  verify.  It is a simple statement that  now possesses four  inline citations to support the claim.  WP:V  does not, as far as I  can read, require that the claim be an exact  quotation from any of the  reference sources.  So what exactly do you  want to see here, Paradoctor?
 * &mdash;   Paine (<font size="2"   color="darkblue"   face="Ariel">Ellsworth's  <font   color="blue" face="Ariel">Climax )  23:05,  25  February 2010 (UTC)
 * "quotation on talk": That's the only tag I found, I will add relevant quotes to the corresponding citation. If this comes up in the future,  I'll make a more appropriately-worded tag.
 * "what exactly do you want to see": Steaphen has implicitly challenged the sources as not  actually supporting the statement. The quotation request serves to  provide direct evidence of what the source says, instead of having to  trust an editor's claim that the source provided really supports the  claim it's attached to. The idea is to prove "book B says A" by saying  '"blah blah A blah blah" is in B on page p'. The former may be difficult  to verify, the latter necessitates only functioning eyes. Also, I don't  have access to one of the sources. Paradoctor  (talk) 00:35, 26 February 2010 (UTC)


 * "Steaphen has implicitly challenged the sources as not actually supporting the  statement" -- It would seem that at least one of you is starting to  understand the nature of facts, and statements concerning them.  But, a  small correction -- I don't implicitly challenge such statements, I  most explicitly challenge them. That was the impetus for mediation.  There is no Reliable Source who fully confirms (citing evidence) the  statement 'we may calculate', beyond the presumptive speculations common  to mathematicians. By all means cite said presumptive speculations and  opinions, but to state as fact, 'we may calculate' is simple wrong, as  I've many times stated. And to ignore the presumptive element in  statements by 'Reliable Sources' who say otherwise is simply bad  science.Steaphen (talk) 01:22,  26 February 2010 (UTC)
 * Analysing reliable sources' arguments is not our job. If a reliable source says "X", we have to use "X".  WP:DUE allows us to omit non-RS, but that's the extent of it.  Whatever the RS say, we report on it. Paradoctor  (talk) 01:42, 26 February 2010 (UTC)

Temporary notice, please read
I had some wetware downtime, that's why I didn't respond earlier. I'm going to comment out anything below not belonging in this section. If any of you does not want this, no problem, just move this into its own section, and remove  this paragraph. Regards, Paradoctor (talk) 21:15, 1 March 2010 (UTC)

It looks like everyone had opportunity to see this. I proceed as announced later this day. Paradoctor (talk) 11:59, 2 March 2010 (UTC)


 * Good, looking forward to seeing Reliable Sources who state that we may calculate, precisely, the actual  movement of runners, arrows etc, at and below the Planck length, through  which runners, arrows and the like, must  travel.Steaphen (talk) 02:03,  26 February 2010 (UTC)


 * Why? the word "precisely" is not used in the claim.  And as I think I mentioned, no one will find any reliable  sources that will say anything scientific about measurements below the  Planck length, because that is the shortest length that "has meaning".   But I'm sure you already knew that.  Suppose the wording were modified  something like the following:  "While mathematics now has formulas that  can be used to calculate with fair accuracy where and when the moving  Achilles would overtake and pass the Tortoise, . . ."?  This is a good  bipolar lead in to the philosophical content that follows the claim in  the article, isn't it? and it also explicitly implies with the term  "fair accuracy" that Planck-length precision is not claimed.
 * &mdash;  Paine (<font size="2"  color="darkblue"  face="Ariel">Ellsworth's  <font  color="blue" face="Ariel">Climax )  04:41, 26  February 2010 (UTC)


 * "Fair accuracy" would suffice in that it conveys a truer description of reality, and does not infer a solution  to the paradoxes. But as for the need for explicit 'precise' statements,  that is exactly what is required whenever there is some inference that  mathematics can be used to precisely solve the paradox of motion (of  physical things). And since no such statements will be forthcoming (at  least not by a reputable Reliable Source), we may conclude with the edit  as suggested ... with mathematics offering 'fair accuracy', and only  fair accuracy.


 * Let that be the end of it.Steaphen (talk) 08:34, 26  February 2010 (UTC)


 * ✅, and thank you very much, Steaphen!
 * &mdash; Paine  (<font size="2" color="darkblue"  face="Ariel">Ellsworth's  <font  color="blue" face="Ariel">Climax )  09:20, 26  February 2010 (UTC)


 * Actually, it is a high-degree of accuracy AND a high-degree of precision, all depending on how accurate  the measurements are and how constant the velocity is. The amount of  uncertainty in calculating how long it will take a radio beam (much  faster than Achilles) to bounce back from a satellite (much further away  than the turtle, but also "moving") is phenomenally low. "Fair" might  satisfy Steaphen, but it is a radical  understatement.--JimWae (talk)  09:34, 26 February 2010 (UTC)


 * (ec). If true consensus is earnestly sought, Jim, then editors must agree to make concessions at  some point.  "Fair" means many things in different contexts.  In this  context its definition is "quite good", and does not in any way imply  inaccuracy or even low accuracy.  Please remember that we are not  claiming anything other than the fact that mathematics can be used to  calculate when and where Achilles would catch and pass the tortoise.   And while "fair accuracy" might seem an understatement to some, others  may yet consider it overstatement.  Perhaps to say "fair accuracy", to  mean "quite good accuracy", would be an acceptable "middle ground"?   That is my hope.
 * &mdash; Paine  (<font size="2" color="darkblue"  face="Ariel">Ellsworth's  <font  color="blue" face="Ariel">Climax )  10:07, 26  February 2010 (UTC)


 * "high-degree', like 'fair-accuracy' is a relative term, in contrast to the exact 1:1 correspondence demanded by  infinite-series, or other mathematical solutions that purport to 'solve'  the paradoxes.
 * Let 'fair accuracy' be the end of it, as agreed.Steaphen (talk) 10:01,  26 February 2010 (UTC)


 * Yes, high-degree (like "fair") is a relative term, not an absolute term. Not being an absolute term, it does  not claim infinite precision nor infinite accuracy & thus satisfies  your objection--JimWae (talk)  18:54, 26 February 2010 (UTC)


 * I hope you are all aware that the current version says that  you can solve  the equations that are defined by Zeno paradox can only be solved with  fair accuracy. It says that the sum of the geometric series is only with  fair accuracy 2.  Paradoctor commented out my  comments, and thus none of them have been addressed.
 * Zeno's paradoxes are not about Newtonian mechanics, which were introduces some  2000 years later. The fact that Quantum mechanices supersedes Newtonian  mechanics is true, but nor relevant for Zeno's paradox. Zeno's paradox  is about the impossibility of supertasks.
 * Zeno's paradoxes are mentioned in Aristoteles Physics, but at that time physics was a  branch of philosophy, not what we currently understand by physics. The  paradox relies on the mathematical property that any rational distance  can be divided in two, rather than on any measured physical property of  the runner, such as his weight, speed, acceleration, size, charge,  spin.  It is not an actual physics experiment with an actual runner and  tortoise to show a certain model of motion is wrong, but a thought  experiment with a mythical runner and an proverbial tortoise to show  that the concept of motion leads to an apparent logical contradiction.
 * Do we have confirming that Zeno describing a physics experiment? Because  the current formulation suggests it does. For such an experiment you can  with fair accuracy measure the positions, and then calculate in a model  exactly - given the numerical accuracy - what the prediction will be,  and you can then measure with some accuracy whether the prediction fits.  If Zeno arguement is however a proof by contradiction, there is no  measurement involved, and all calculations are  exact.Ansgarf (talk) 01:01, 27  February 2010 (UTC)


 * Ansgar, "and then calculate in a model exactly" -- astrology does this as well. See below   - if you can cite a  reliable Reliable Source, someone who correlates reality with theory,  please enlighten us.


 * Otherwise, as agreed, let 'fair-accuracy' be the end of it (no pun intended .. but not a bad  one, now that I think about it. Quantum superpositioning working for me  again. :)Steaphen (talk) 03:45,  27 February 2010 (UTC)

Yes, and then 'calculate in a model exactly. And them confirm -with some accuracy- the predictions by measurement. There is nothing unscientific by doing first the math, and then the experiments. The only commonality with astrology is that they use math. But your tax officer uses math too. And quantum mechanics too. What inaccuracy do you expect when calculating within a model, other than numerical inaccuracy? That the Newtonian model is incorrect is true, but that is not a problem of the math involved. Do you have a reference that Zeno was describing a real tortoise and a real runner? Do you have a reference that he was describing an actual experiment, and  that it is not an "example of a method of proof called reductio ad  absurdum". Because that is what the article currently still says. Ansgarf (talk) 04:11, 27 February 2010 (UTC)


 * Two people agreed on something, and now 2 other people think the agreement came too soon. Case is not  closed--JimWae (talk) 04:10, 27  February 2010 (UTC)
 * I am not opposed to mentioning that the Newtownian model that is usually used when Zeno paradoxes are discussed  is only an approximation of reality. If we find a source that links this  to Zeno. Otherwise this is a fact to be discussed here.
 * As it is we should remove the cited references, because they all state that you can do the math, and not just  approximately. Ansgarf (talk)  04:18, 27 February 2010 (UTC)


 * (Ansgar) What (sic) math are you referring too? What connection with reality does that math actually  entail, beyond unfounded speculation and supposition (like astrology)?
 * Math deals with the fact that the sum of the geometric series is exactly 2.  And not approximately 2. There is nothing unfounded about that theorem.  Ansgarf (talk) 06:33, 27 February  2010 (UTC)

'''
 * Again, what Reliable Source can you cite that confirms geometric series can be applied to map or explain the minutia  of movement of physical things, such as Zeno's arrow, runner, tortoise?  If no sources, then go away please.
 * The geometric series converges regardless of whether it used to in a Newtonian model, a  Quantum model, or Financial forecasting. The model might be wrong, but  the series converges nevertheless. See Kreyzigs calculus book.  Ansgarf (talk) 08:50, 27 February  2010 (UTC)
 * (JimWae, Angar et al) If any of you have a reliable Reliable Source who can confirm beyond 'fair-accuracy' or even  'high-accuracy the efficacy and experimental validity of any  mathematics in relation to the minutia of movement of physical things  (e.g. arrows, hares etc.) then by all means present them, otherwise,  as agreed, let 'fair-accuracy' be the end of  it.'Steaphen (talk) 04:41, 27  February 2010 (UTC)


 * None of the refs attached to the sentence mention "fair accuracy" either -- or any degree of accuracy at all.  http://www.lpi.usra.edu/lunar/missions/apollo/apollo_11/experiments/lrr/  gives a error margin in measuring the distance to the moon of less than  0.00000000078%, with the moon receding 3.8 cm/yr. With lesser distances  (and a stronger return signal) the margin of error would be even less.  To call such precision "fair" is beyond an understatement. Some other  language needs to be found. There are many terms that mean something  less than absolutely infinite precision - (it was only you who ever  wanted to specify a degree of precision or accuracy - no claim about  either was previously made at all). I also must note (in line with  Ansgar's comments) that we need to be careful about whether we are  talking about the model (in which case there can be infinite  "precision"), the application of the model to reality (in which case  "accuracy" is the appropriate term), or the degree of uncertainty in  measurement (again "precision")--JimWae (talk) 05:22, 27 February 2010 (UTC)


 * It seems to me that we are squabbling about 'fair' as opposed to 'high' ... fair  enough. Let 'high-accuracy' then be the compromise that is agreed!   Agree?Steaphen (talk) 05:27, 27  February 2010 (UTC)


 * Well that's a refreshing exchange! We still need to hear from Ansgar, tho'--JimWae (talk) 05:38, 27 February 2010 (UTC)


 * I'll take that as a 'yes, agree' to use 'high-accuracy' in replace of  'fair-accuracy'. I hope you appreciate this translates as "we may  calculate with high accuracy ...", and in no way confirms we may  calculate precisely, at and below the Planck length. As for models, you  may all debate (enmasse :) until the cows come home, but no one is  justified in linking or correlating those models with valid solutions to  Zeno's Paradoxes, at least not beyond 'high-accuracy' as agreed. Thus,  the paradoxes have not been "solved" either mathematically or  philosophically. They have however been approximately solved, to  high-accuracy, no more or less than the earth is approximately flat  (within certain limits). Steaphen (talk) 05:51, 27 February 2010 (UTC)
 * The paradox is not about an actual race between a runner and a tortoise. If  it were an actual race in ancient Greece, I would be happy to say that a  runner, who is likely called Achilles, was running to a position that  was with high accuracy the same position where an unnamed tortoise or  turtle approximately was the start of the race, but that the turtle or  tortoise had moved on in the meanwhile.
 * The article gives a description of the paradox in terms of the geometric series. This is a  mathematical model. It mentions that the paradox is a reductio ad  absurdum, thus a logical argument.  Nowhere does the article, or the  historical sources claim that it is an actual race.
 * I am not opposed to mention that the "problem of motion" also exist in physics.  Peter Lynds for example states that while mathematics gives the exact  solution to the paradox, it fails to explain how motion is possible at  all in an quantum physical model.
 * So, I propose not to change to fairly to highly accurate for positions that happen in  a model. My proposal is to make clear when we are talking about the  runner and tortoise from the paradox, and when we talk about the  movement of actual bodies. The references talk about the tortoise and  the runner in the model of the paradox, and for these they give ways to  compute the intersection of their trajectories. Statements of  mathematics should be unqualified -no "highly" or "fairly" since they  are exact.
 * When it comes to measuring actual positions, I don't oppose to use a "within the limits of QM", or a "highly accurate".  And if you want to mention that the model of motion that is commonly  used to describe the paradox, namely Newtonian mechanics, is only  approximately correct and has been superseded by QM, feel free to do it.  Ansgarf (talk) 06:33, 27  February 2010 (UTC)


 * Right, so let me see if I understand you correctly. Zeno's Paradoxes is not about the actual movement of  runners. It's about some hypothetical runner, and some hypothetical  tortoise? Correct? Right then, what qualities do these hypothetical  runners possess? Are they in any way related to real runners, that  actually run? If not, let's make up whatever we like .. that they can  fly, and jump through hyper-space, whatever.


 * So, help me out here, what planet are we now on?


 * It seems to me Ansgar is being deliberately obstructive. Do we now call/arbitrate/agree  to disallow further comments by Ansgar? Besides Ansgar is there anyone  else who has problems with 'high-accuracy' in relation to Zeno's  Paradoxes?Steaphen (talk)  06:53, 27 February 2010 (UTC)
 * I have argued for a while that the paradox is about a hypothetical race between a mythical runner  Achilles and a proverbial tortoise. Yes. Do we have sources that tell us  when the race happened?
 * You ask me what qualities the hypothetical runners posses. A good question. What quantified physical  quality of the actual runner is specified in Zeno's paradox? None. Not  his height, not even his speed, not his weight. Or what do the sources  say?
 * Also, what quantified physical quality does Zeno use in his argument? Does he argue that Zeno is too heavy? Too long? No,  none of these. To the contrary, Zeno's argument hinges on the property  that in-between any two rational numbers is another. This is not a  physical property of the runner but a mathematical property of the  rational numbers. The above statements are based on the description  given by Aristotle. Ansgarf (talk) 08:39, 27 February 2010 (UTC)
 * My proposal remains: Let's distinguish clearly when we talk about the  runner and tortoise from the paradox (the map), or when talk about  physics theories of motion of physical bodies (the  ground).Ansgarf (talk) 08:39, 27  February 2010 (UTC)

I suggest that Ansgar's reply shows that he is determined to be obstructive. To suggest that for Zeno to believe runners, hypothetical or otherwise did not share common physical  characteristics is the height of stupidity. Runners, tortoises and arrows are physical things with quantifiable physical characteristics,  the specific values of which are largely irrelevant when studying the  finer movement of physical objects. Movement through the Planck length is independent of object size ... the object, irrespective of size and  weight, moves incrementally, through increments that must include the  Planck length and shorter. Hence the validity and correctness of the 'high-accuracy' term in relation to mathematical theories relating to  Zeno's Paradoxes.

My original purpose here was to install some discipline in the treatment of Zeno's Paradoxes. Ansgar, either desist with this nonsense, or arbitration will be requested to restrict your  contributions.Steaphen (talk)  10:39, 27 February 2010 (UTC)
 * I didn't suggest that Zeno didn't believe that real runners have some common characteristics, but he  didn't mention a single of these characteristics in the paradox. I am  still waiting for your reliable sources to the contrary. He mention a  characteristic of rational numbers though. The source is Aristotle, but I  mentioned that before.
 * As an aside, the size of and weight of an object does matter in quantum mechanics more than in a classical model,  especially since there are smallest distances and energy quanta. How  exact the position can be determined is for example influenced by its  momentum. You might recall Broglie wavelength. I won't comment much on  your realistic and local interpretation of what happens between Planck  lengths, but I just want to point out that it is not the only  Interpreation of Quantum Mechanics. Without sources that relate them  to Zeno there is no need to mention them in the article.  Ansgarf (talk) 12:23, 27 February  2010 (UTC)


 * According to historical records, Zeno didn't mention calculus. The interpretations of quantum theory are not in  contention. Nor are other aspects of quantum theory. Nor is calculus,  beyond it providing 'high-accuracy' in 'approximately solving' Zeno's  Paradoxes.


 * I move that the content reflect the agreed 'high-accuracy' in relation to mathematics providing 'high-accuracy'  solutions to Zeno's Paradoxes. All agreed? Steaphen  (talk) 12:55, 27 February 2010 (UTC)
 * Zeno did mention that in the diochotmy paradox that moves half the distance.  This gives a geometric series. We can skip calculus, if that is the  problem. The discovery of the convergence of geometric series predates  the invention of calculus. Also, the fact that the sources that gave I  may be inadeaquate doesn't excuse that you haven't provided any for your  claims. Ansgarf (talk) 14:59, 27  February 2010 (UTC)

A comment by Steaphen
To be succinct: stick to good scientific, journalistic principles, and the  finer detail of what to say, or write will take care of itself. Don't state things which are not fact, as fact. E.g. 'using calculus we can calculate when Achilles overtakes the tortoise.' According to the  evidence that statement is demonstrably incorrect. If you can cite a Reliable Source who says you can, precisely, including at and below the  Planck length (required by infinite-series solutions, or any  mathematical solution that infers a deterministic, 1:1 correspondence  between theory and fact) then great, please include. Nobel Prize to the Source, guaranteed. And good on you for bringing that Source to light.

The rest, as I said, will take care of itself. It has been the lack of scientific method/principle that has been the impetus for mediation.

And Ansgar, you do test me, by misquoting me - I did not write or infer  that the use mathematics is wrong. For your benefit, paraphrasing earlier - While Newtonian mechanics, and the mathematics upon which it  is based is quite useful at calculating the when and where of things,  when considered in detail, such methods fail to produce precise,  predictable results. That is what the experimental evidence has revealed, and continues to reveal. Thus, any statement which directly states or infers that "we can calculate" is, when considered in detail,  incorrect, in the same manner as it would be incorrect to say that the  Earth is flat. Within certain crude approximation (e.g. my living room floor) it is indeed flat, but in more global, precise terms, such a  statement is demonstrably wrong.

In conclusion, by all means state the use of calculus or whatever, and its historical use, but in no  way can you state that 'we can calculate when and where Achilles  overtakes the tortoise' because a) no reliable source has said as much,  in detail, including at and below the Planck length, and b) the evidence  confirms you can't make such statements with any validity or that it  correlates with actual reality.

By all means! report the historical beliefs and opinions and statements of mathematicians who  believe whatever, but, statements such as "we can calculate when  Achilles overtakes the tortoise" are not facts, in that they cannot be  supported by evidence, and thus remain speculative opinions. Cite one competent Reliable Source who confirms otherwise, and I'll be applauding  a Nobel Prize well deserved.Steaphen (talk) 08:29, 25 February 2010 (UTC)


 * "E.g. 'using calculus we can calculate when Achilles overtakes the  tortoise.'  According to the evidence that statement is demonstrably  incorrect.":  Evidence, in here, consists of citations to WP:RS. Where are they?  Paradoctor (talk) 09:08, 25  February 2010 (UTC)


 * It is the use of statements as indicated e.g. 'we can calculate' that is in contention, not whether the converse  is true or not. This is a talk page. There is no need whatsoever for any  Reliable Source to confirm any of my statements or beliefs as they are  not on the main article page. I hope this clarifies matters for you.  Again, it is the lack of principle that is being called into question,  regards statements on the main article page.Steaphen  (talk) 09:12, 25 February 2010 (UTC)


 * Are you proposing any specific changes to the article? If not, then I will delete this section per WP:TALKNO and WP:DISRUPTIVE, as it  constitutes "discussing the topic", and is demotivating to me, possibly  for others. You can discuss on user talkpages to your heart's desire, I  won't mind the least. Please respect that this talkpage is a workshop,  not a forum. Paradoctor (talk) 09:36, 25 February 2010 (UTC)


 * (Note: Steaphen reacted by adding specific proposals, which I moved up one  section. That's the reason I have not deleted this section.  Paradoctor (talk) 13:50, 25  February 2010 (UTC))

✅

Summarizing the Lee paper?
Ansgarf recently added a paper by Harold Lee, whose points I think should be explicated in this article. Basically Lee says that Zeno's premise is wrong. Specifically, Zeno seems to have assumed that space is merely comprised of rational numbers, which (as we now  know) are inadequate to construct a continuum. In a world of only rational numbers, Zeno has showed that motion is impossible ("Thus,  within the terms of Zeno's analysis, there can be no motion"). So obviously, motion is not a process that merely constitutes successive  additions of rational numbers. Lee then says that if we assume that space is comprised of real numbers, there is no paradox. An interesting point, and relevant to the topic of the article. Would anyone mind if I wrote a summary of the paper in the article? Gabbe (talk) 13:54, 24 February 2010 (UTC)


 * I'd like to see such a summary. While it may be relevant to "Zeno in literature", I cannot see how that in any way will "solve"  the paradoxes. There is still no "next number" nor "next step" and no  "last number in the sequence" nor "last step" to begin finishing. Zeno's  paradoxes can confine itself to rational numbers (and distances,  whatever that may mean) because that is all that is needed to present  the paradoxes.  Whether he thought there were ONLY rational numbers  (unlikely) or that space can ONLY be divided into distances with a  rational length (whatever that may mean) would not matter. But, I'd like  to see the summary.--JimWae (talk)  19:36, 24 February 2010 (UTC)
 * Exactly. The paper doesn't "solve" the paradoxes, it merely makes explicit some of the implicit assumptions  necessary for the paradoxes to be paradoxes. Gabbe  (talk) 06:48, 25 February 2010 (UTC)


 * (JimWae) 'how that in any way will "solve" the paradoxes': Um, not our job. What constitutes a solution is something the experts are  disagreeing  about, so we report on who said what. Just thought I'd  mention that. ;)
 * (JimWae) "may be relevant to "Zeno in literature"": Disagree, it makes an argument relevant to Zeno's  paradoxes, rather than using it as a plot device in a narrative.
 * (Gabbe): "summary": Yes, please. Paradoctor (talk) 08:25, 25 February 2010 (UTC)


 * I have now seen the first page of the paper & see that it was not a work  of science fiction (as some of the refs have been). Mind also  published a short response to Lee's 1965 article (  http://www.jstor.org/pss/2252470 ). I see now that the article is being  used only to support the claim that the mathematical issues have been  resolved. I am fine saying there are no mathematical issues in finding  the sum, but from what I can see this article may claim to do more than  that.  Btw, are scholars still citing or discussing Lee's work?  I must  add that motion is not "a process of adding" any type of numbers.
 * I also would prefer, instead of "While mathematics can be used to  calculate where and when the moving Achilles will overtake the  Tortoise" if the word would replaced "will", thus making it less  of a claim about reality and more a claim about the model of reality.  --JimWae (talk) 09:34, 25 February  2010 (UTC)


 * Lee's paper is listed in Salmon's bibliography, that's sufficient to regard it as relevant to us, IMO. I have no  objection to the proposed change in wording. It might be a good idea to  check the phrasing against what is used in the literature.  Paradoctor (talk) 09:48, 25  February 2010 (UTC)


 * "Will" or "would", no matter, because it won't change Steaphen's argument about "mathematics can . . .  calculate".  Maybe something like "While mathematics now has formulas  that can be used to calculate with fair accuracy where and when the  moving Achilles would overtake and pass the Tortoise, . . ." might get  the article closer to what Steaphen is striving toward?
 * &mdash;  Paine (<font size="2"  color="darkblue"  face="Ariel">Ellsworth's  <font  color="blue" face="Ariel">Climax )  04:32, 26  February 2010 (UTC)


 * I do see 2 other later works by Lee in Salmon's bibliography--JimWae (talk) 10:00, 25 February 2010 (UTC)
 * That would be the 2001 edition. Fricken preview limit. ;)  Paradoctor (talk) 10:09, 25  February 2010 (UTC)

In regards to this material by Lee, in which he refers to Dedekind and Cantor: I was unaware that they were  physicists who correlated theory with reality. Perhaps you can advise where within the article those correlations are confirmed sufficiently  to be relevant for this subject. Beyond the usual assumptions and speculations made by mathematicians, many of which are quite clever and,  some might say, even beautiful, how is any of this relevant to the  mediation request to disallow speculation that is unfounded and  unsupported by fact? Have any of you understand anything regarding the scientific method?Steaphen (talk) 15:36, 26 February 2010 (UTC)

On the subject of reliable Reliable Sources
'Shifting deckchairs on the Titanic' is perhaps an apt metaphor for some of the discussion on  this Talk page regarding Zeno's Paradoxes.

Zeno's Paradoxes, despite claims to the contrary is first and foremost about the inquiry  (historically, by Zeno of Elea) into the actual detail of how things  (arrows, runners et al) move, and that inquiry, based on certain  assumptions, elicited the paradoxes as historically recorded. But Zeno, as historically recorded, started first with the inquiry into the  movement of physical things.

We can confidently expect that any discussion, or contribution that is not centred on the detail of the  movement of physical things - as historically queried by Zeno - is  doomed to sink into irrelevance, at least in regards to the subject of  "Zeno's Paradoxes".

Any so-called Reliable Sources who DOES NOT explain or cover how things move, in the minutia, and how that  description of movement is correlated with fact, can be deemed  'unreliable'.

If anyone reading this page can explain how any such "Reliable' Source should be deemed reliable when that contribution  does not correlate theory with fact, please present arguments below  (not in) the list of issues provided.

This in effect is asking editors to explain why we should ignore or reject good  scientific/journalistic principles (i.e. the scientific method), and  simple 'common sense'.

Is it common sense to talk about theories of physical movement when those theories cannot be substantiated by the  observation of actual physical movement? Moreover, does it follow good scientific principles to believe some theory (physical continuity of  movement) when that theory flies in the face of experimental data (that  in the minutia, 'movement is discontinuous')? If theories that cannot be substantiated in fact, or even strongly correlated with experimental  data are deemed reliable, upon what credible basis are other theories  that also are not strongly correlated with fact, rejected? (e.g. by using astrology, that "at such and such an hour, due to Mars being in  Pluto, Achilles will overtake the tortoise")?

In conclusion, if the Reliable Source does not correlate or in some manner substantiate  theory with observable fact, I propose that it be deemed 'unreliable'  and therefore rejected.

However, this does not disallow 'Reliable Sources' who comment on the historical beliefs concerning this  subject, and can include references to any and all mathematical  theories that purport to 'solve' the paradoxes. But the preface to any such inclusion needs the "mathematicians believed" qualification ... and  a postscript "but those theories were not and have not been  substantiated in fact."

Some examples of phrases and sentences that therefore need to be reworked or deleted entirely, include:

<ol><li><font color="#990000">"Zeno's paradoxes were a major problem for ancient  and medieval philosophers. More modern calculus has solved the  mathematical aspects of the paradox". Where is the evidence that modern calculus solves the paradox of movement? What experimental data that includes movement through all increments can be reliably  sourced for this statement?

<li> <li><font color="#990000">"while many philosophers still hesitate to say that  other aspects of the paradoxes are completely  solved" <ol> Which philosophers  'hesitate'? And what about those that don't hesitate in the least, but state categorically calculus does not solve the  paradoxes?</li> What part of the paradoxes are  even partially solved? What reliable "Reliable Sources" support the speculation that they are even "partially solved". Perhaps the reworked statement should include the 'fair accuracy' offered by  (<font size="2"  color="darkblue"   face="Ariel">Ellsworth ), which ties 'fair accuracy' to  any such 'partial solutions'.</li></ol></li> </li> </li> <li>What reliable Reliable Sources confirm that the paradoxes are solved beyond 'fair  accuracy', and what is the specific experimental evidence supporting  that claim?</li> <li><font color="#990000">"Modern calculus achieves the same result, using more rigorous methods (see  convergent series, where the "reciprocals of powers of 2" series,  equivalent to the Dichotomy Paradoxmore" - what  experimental data supports the linking or equivalence of "reciprocals of  powers of 2" to actual physical movement? What reliable Reliable  Sources confirm this equivalence, and what   experimental evidence do  they cite?</li> <li><font color="#990000">"Another  proposed solution[citation needed] is to question the assumption  inherent in Zeno's paradox, which is that between any two different  points in space (or time), there is always another point. Without this  assumption there are only a finite number of distances between two  points, hence the infinite sequence of events is avoided, and the  paradox resolved." How specifically does this assumption  remove the paradox of movement? If there are only a finite number of distances between two points, how do we move between those points? What is the ground, or transport mechanism that 'jumps' us from point A to B?  Furthermore, what reliable Reliable Source confirms paradox is avoided  if there are only a finite number of points?</li> <li><font color="#990000">"This effect is usually called the "quantum Zeno  effect" as it is strongly reminiscent of (but not fundamentally related  to) Zeno's arrow paradox." How is the evolution of quantum systems (which according to physicists comprises  the entire universe, and all of us - hence Schrodinger's cat, Many  Worlds Interpretation etc.), not fundamentally related to Zeno's  Paradoxes? What reliable Reliable Source confirms a fundamental disconnect between quantum and everyday systems that is inferred and  required by 'but not fundamentally related to'? Proposed change (in the least): simple remove the unfounded, and unsupported  speculation 'but not fundamentally related to'</li> <li> ...more, soon. Steaphen (talk) 03:20, 27 February 2010 (UTC)</li> </ol>
 * Proposed change: " More modern calculus has  solved, with high accuracy, the mathematical aspects of the  paradox. " * Done.
 * Proposed change: remove the prejudicial 'hesitate' and replace with "Many philosophers argue that,  beyond any high-accuracy solutions, the paradoxes remain  unsolved."
 * update: 'fair accuracy' changed to 'high accuracy' pursuit to the above  agreement with JimWae. Steaphen (talk) 13:09, 27 February 2010 (UTC)

New proposal by Ansgarf
I propose to make explicit when we talk about the object described by the paradox. The does justice to the fact that the paradox is introduced as a proof by contradiction, thus a  logical argument. It also does justice to the description of the paradox, in the article, and in all classic sources, that describe e.g.  the dichotomy paradox as geometric series. This however also addresses the fact that when the model that is used in the paradox is applied to  real bodies, you have the problem with measurements and uncertainty.

In particular I propose the following: While mathematics can be used to calculate, given the above description, where  and when the moving Achilles will overtake the Tortoise of Zeno's  paradox  some philosophers say that the mathematics does not address the  central point in Zeno's argument, and that solving the mathematical  issues does not solve every issue the paradoxes present. This would also take into account that the references actually do say that  you can calculate it. None says that you can only calculate it approximately. For that claim we would need new sources. No need for change. You can solve mathematical aspects with mathematics. The "hesitate" is already removed. It was indeed too vague. I see no further need for change, but wouldn't object to change it, based on Steaphen proposal to:
 * "Modern calculus has solved the mathematical aspects of the paradox"
 * "while many philosophers still hesitate to say that other aspects of the paradoxes are completely solved"
 * Many philosophers argue that beyond mathematical solutions the paradoxes  remain unsolved.

Most request for sources have been dealt with. And finally, mathematical theorems should not be qualified with "highly accurate" since they are mathematically precise and not approximate. Ansgarf (talk) 15:21, 27 February 2010 (UTC)


 * Provide a reliable source who links the inventions of mind you speak of  (mathematics) with the actual movement of runners, including at and  below the Planck length. Otherwise, your material is blatant and  inexcusable assumptive, illogical prejudiced opinion, and all sections  with any inference that infers mathematics solves anything will be  removed! Then we'll have a case for arbitration to intervene. You are  being deliberately obstructive. Others have agreed to 'high-accuracy'  beyond which there is absolutely no justification for stating  mathematics solves anything. That is the fact of the matter, as  confirmed by experimental evidence.Steaphen (talk) 22:29, 27 February 2010 (UTC)
 * You made two contentious edits. First, the other editor have not responded to your  request to say that "calculus solves mathematical aspects" only with  "high accuracy". Sources that show that you can solve these problems  precisely are books on numerical analysis.  I include a seminal paper by  Kung on the topic, and changed sentence accordingly. I would be happy  to use my proposal Many philosophers argue that beyond mathematical  solutions the paradoxes remain unsolved. as a compromise.
 * With respect to whether we can compute positions with high precision, I  haven't seen that anybody has disagreed with my proposed formulation,  but you. We should simply wait for others to have a say. Jim wasn't  opposed to "high accuracy", but suggested to wait for my input. So, you  lets wait for the others. But we can keep the sentence as it it is for a  while. No need to hurry.
 * The other changes seem ok with me, but I moved the POV tag to the top. By the way. The current sources say that  Weierstrass has solved the mathematical problems of the paradoxes, but  it seems you argue that this is POV. Do you have sources that claim the  opposite?Ansgarf (talk) 23:29, 27  February 2010 (UTC)
 * It is about time to address the point that you cannot calculate "distances". I agree, you can only walk distances,  or see distances. Distances are not numbers, and you can only compute  with numbers. You can compute the length of a distance though. And you  can measure length of distances. But this is a problem of Ontology  not of Quantum Mechanics. Ansgarf (talk) 23:47, 27 February 2010 (UTC)


 * Your edit has been reverted. The Reliable Source removed, as it did not mention  movement through Planck and shorter increments, and thus was  speculative.Steaphen (talk)  00:03, 28 February 2010 (UTC)
 * Kungs paper wasn't speculative, it showed how to exactly compute reciprocals of powers. And I thought that  was in contention. As far as I know, Kungs paper is correct. But it did  not mention Zeno explicitly, I grant you that. The Lynds paper says  "The way in which calculus is often used to solve Achilles and the  Tortoise and the Dichotomy through the summation of an infinite series  by employing the mathematical techniques developed by Cauchy,  Weierstrass, Dedekind and Cantor, certainly provides the correct answer  in a strictly mathematical sense by giving up the desired numbers at the  end of calculation." He does say "correct answer" not "highly  accurate answer".
 * The Boyer book mentions Zeno a few dozen of times, and the Zhang paper explicitly addresses how to model systems  with Zeno behaviour, and how to solve them. I added those references and  changed the article accordingly. Ansgarf (talk) 02:01, 28 February 2010 (UTC)
 * With respect to the the question if there are sources who link mathematics with the  actual movement of objects. Sure I can provide you such sources. In  Relativity: The Special and General Theory by Albert Einstein    it is for example postulated that $$e = m c^2$$.  This relates the mathematical theory of multiplication on the field of  the real numbers to energy, mass and the speed of light. If you are  interested in a publication that links mathematic to quantum mechanics,  including at and below the Planck length, look at Schroedingers 1926  paper An Undulatory Theory of the Mechanics of Atoms and Molecules.  In it he gives mathematical equations and relies on the correctness of  modern calculus to describe according to the abstract "wave-length,  macro-mechanical and micro-mechanical problems", then the "The  wave-equation and its application to the hydrogen atom", but also "Other  problems; intensity of emitted light".
 * I am happy to provide you more reliable sources that link mathematics to the  movement of physical things. Kreysigs "Advanced Engineering Mathematics"  is a start, Boyers "The History of the Calculus and Its Conceptual  Development" even mentions Zeno a few doaen of times, then there and the  majority of the paper on physics and engineering conferences, like the  Americam Control Conference (ACC) from 1982 to 2009  . See the  2009 proceedings as example  .Ansgarf  (talk) 02:01, 28 February 2010 (UTC)
 * By the way I am still waiting for sources that confirm that Zeno was  describing a physics experiment. Because your requests for more sources  are off-topic if he didn't.Ansgarf (talk) 02:04, 28 February 2010 (UTC)


 * Your edit has been reverted, as it did not provide a Reliable Source covering  movement through Planck and shorter increments, and thus was  speculative.
 * Do you have a source that mentions Planck length as part part of the "mathematical aspects" of Zeno's paradoxes? You  have not given one, even though I asked repeatedly it. Please provide  one before your next revert or edit.
 * Furthermore, I wonder which of the sources did not talk about infinitesimally small numbers?  The Zhang reference certainly did, as did the Lynds paper, and also  Boyer book does so. Could you please you respond to my comments, rather  than ignore them. Ansgarf (talk)  02:22, 28 February 2010 (UTC)


 * Your edit was reverted, because it did not provide a Reliable Source covering movement through  Planck and shorter increments, and thus involved speculation beyond the  'high-accuracy' of mathematical solutions, as agreed above by JimWae et  al. The onus is upon those making assertions to back them up with  credible Reliable Sources who address the issue of movement through  increments, including at and below the Planck  length.Steaphen (talk) 02:26,  28 February 2010 (UTC)
 * As far Jim did not agree to changing that sentence, and suggested to wait for my input. I haven't seen a  response of any other editor to my proposal, and your response to it. It  would be appropriate to wait for others to  respond.Ansgarf (talk) 02:36, 28  February 2010 (UTC)

My proposal remains: there is no need to qualify the the use of calculus for mathematical problems with "highly  accurate". You can solve mathematical aspects with mathematics. I propose to either I haven't had any reply other than the unsourced statement that the mathematical  aspects mention explicitly motion of objects at or below Planck length. Neither has anyone provided a source that Zeno paradox is not a paradox of  mathematics and logic. Even though I asked for it. The sources that I provided all deal with infinitesimally small numbers, and all of them  address Zeno's paradoxes. Ansgarf (talk) 02:36, 28 February 2010 (UTC)
 * Revert to the old version
 * Omit the "highly accurate"
 * Or change it to Many philosophers argue that beyond mathematical solutions the paradoxes remain unsolved.


 * The onus is upon those making assertions to back them up with credible Reliable  Sources who address the issue of movement through all increments,  including those at and below the Planck length. Since none have been  provided, the 'high-accuracy' is both valid and  appropriate.Steaphen (talk)  02:41, 28 February 2010 (UTC)
 * The Zhang paper deal with all increments. See section 4 Zeno Hybrid Automata and definition 11.
 * The Lee paper does deal with all increments. See page 4.
 * The Boyer book does deal with all increments. See for example pages 7, 39, 219,  257 or  259.
 * Where it the source that supports the view that mathematical aspects are about motion at or below Planck length?  Ansgarf (talk) 02:54, 28 February  2010 (UTC)


 * Are you asserting that physical things (runners, arrows etc) DO NOT move through increments of length equal or shorter to  the Planck length? If they do, what Reliable Sources confirm the  mathematics that maps objects through those scales? What Reliable  Sources provide mathematics that provides exact detail of how physical  things move and is supported by the experimental evidence? Since you  have not provided any sources, the 'high-accuracy' remains valid and  appropriate.Steaphen (talk)  03:04, 28 February 2010 (UTC)
 * I presented papers that deal with all increments as requested, and these assume that motion happens at all  increments. Whether I personally believe that in physical reality  objects move through all increments, or discretely is not really  relevant. FYI, I am agnostic about this, because you cannot say much  about what happens below planck length. That would be speculative.
 * I assert however that the "mathematical aspects" of Zeno paradoxes the do  not hinge on physical entities such as the Planck length, but on  mathematical properties of the rational numbers. The sources on the  paradox that I provided did not mention that Zeno was discussing  explicitly motion at or below Planck length, and I am asking you to  provide such a reference. Ansgarf (talk) 03:13, 28 February 2010 (UTC)


 * re your "and these assume that motion happens at all increments." This is an  encyclopedia and does not condone or allow speculative assumptions. What  Reliable Source (physicist) can you cite who provides exact  mathematical results for movement of objects through all scales  (including those at and below the Planck length)?


 * The options are, in regards to the relevance and efficacy of mathematics in  detailing and predicting the movement of physical things - 1. zero (0%)  (obviously incorrect, due to Newtons equations etc.), 2. >0%, but  <100% = varying degrees of accuracy and relevance (e.g. poor, fair,  high etc), or 3. 100% absolutely accurate (your option). You have not  provided any sources confirming option 3, that mathematics exactly and  perfectly maps the movement of objects with perfect certainty through  all scales of increment. Failing any sources, the 'high-accuracy' as  agreed by JimWae et al, remains appropriate and  validSteaphen (talk) 03:39, 28  February 2010 (UTC)
 * It wasn't my assumption, it was your question if they did assume that motion happens at all increments and I  confirmed this. It appears that these sources consider all increments.
 * I agree that you cannot compute "physical distances", you can only  compute numbers. You can measure "physical distances" and compare then  with high accuracy to numbers. This is an ontological problem, not a  problem of Planck length. The mathematical aspects however involve only  numbers and mathematical relations (that's why they are called  mathematical). And there is no ontological problem to compute exact  solutions for mathematical problems. And I have provide ample sources  that confirm that you can.
 * I asked you repeatedly to give a source that states that the mathematical aspects hinge for example on  the Planck length. You haven't given one, not even tried as far as I can  tell. With some certainty I am asserting that you do not have a source  that confirms that the mathematical aspects of Zeno's argument depend on  the Planck length.
 * However, I would suggest to wait for feedback of others to my proposal, or to come up with their own  proposal. How does that sound?Ansgarf (talk) 03:48, 28 February 2010 (UTC)


 * As stated above, the onus is upon those making assertions to back them up  with credible Reliable Sources who address the issue of movement of  Zeno's runner, arrow and tortoise through all increments, including  those at and below the Planck length. Upon what basis do you reject  consideration of movement through and below the Planck length? What  reliable sources confirm the validity of your rejection? Since no  credible reliable sources confirm your prejudice against such  considerations, the 'high-accuracy' remains both valid and  appropriate, until confirmed otherwise.
 * Let this be the end of it, subject to amendment if confirmed otherwise.  Steaphen (talk) 04:03, 28  February 2010 (UTC)
 * The basis for my rejection to consider "movement through and below Planck length" explicitly, is that  none of the sources confirm that the paradoxes, let alone the  mathematical aspects thereof, consider "movement through and below  Planck length" explicitly. The main source for this rejection is  Aristotle's description of the paradoxes.
 * Does this answer your question? Do you have sources for the contrary?
 * Also I am not opposed to give the other a chance to voice their views, and  maybe help to resolve the our impasse. Ansgarf (talk) 04:07, 28 February 2010 (UTC)


 * Aristotle also did not explicitly mention runners being able to fly, or  jump through hyperspace. Are you suggesting that you need Reliable  Sources confirming runners do not fly, or are unable to run at 40,000  kms/hour, or ... or ... or  ....
 * The onus is upon those making assertions to back them up with credible Reliable Sources who  address the issue of movement of Zeno's runner, arrow and tortoise  through all increments, including those at and below the Planck length.  Upon what basis do you reject consideration of movement through and  below the Planck length? What reliable sources confirm the validity of  your rejection? Since no credible reliable sources confirm your  prejudice against such considerations, the 'high-accuracy' remains  both valid and appropriate, until confirmed  otherwise.Steaphen (talk)  04:20, 28 February 2010 (UTC)
 * Aristotle did explicitly mention runners and running, but he did indeed not mention  explicitly that they fly or jump through hyperspace. Therefore I  think we should neither cover explictly the cases of a flying  Achilles, or an Achilles that moves through hyperspace. And similarly I  also reject to consider explicitly movement through and below the  Planck length. So, neither hyperspace, nor flying, nor Planck length.  Unless you have sources.
 * The "highly accurate" is inappropriate, since none of the references uses "highly accurate" or  mentions that the mathematical solution to a mathematical problem is  approximate. They all mention that you can solve the mathematical  aspects, without any qualification such as "highly accurate", and some  even state that you can do so exactly. All say e.g. that the sum of the  geometric series is 2, and not with "high accuracy 2". Or did I overlook  that any of the sources mentions that you can't solve the mathematical  aspects mathematically exact? Ansgarf (talk) 06:22, 28 February 2010 (UTC)
 * BTW: I am really happy to wait for others to give their input.  Ansgarf (talk) 06:22, 28 February  2010 (UTC)


 * Aristotle also didn't mention calculus. Let's disallow everything that Aristotle did not mention, as being a  valid treatment of contemporary knowledge. Right. Which planet are we on  again.


 * Re your "And similarly I also reject to consider explicitly movement through and below the Planck length." --  <span style="color: rgb(153, 0, 0); font-weight: bold;">Auchtung  all lookenpeepers! Hands up all those who also reject  considering movement through and below the Planck length, and that we  only need to close our eyes and minds to any such consideration? Any  half-competent thinkers agree? What about some failed physicists, who  wanted to be but couldn't quite get there, do you agree? How about  anyone, other than Ansgar? Don't think too hard, tho', tis easy to  schnappen der springenwork and poppen das fusen.


 * As before, the onus is upon those making assertions to back them up with  credible Reliable Sources who address the issue of movement of Zeno's  runner, arrow and tortoise through all increments, including those at  and below the Planck length. Upon what basis do you reject consideration  of movement through and below the Planck length? What reliable sources  confirm the validity of your rejection? Until shown otherwise the  'high-accuracy' remains both valid and  appropriate.Steaphen (talk)  07:07, 28 February 2010 (UTC)
 * This response was to your previous, less provocative  answer.
 * You are right that Aristotle didn't mention calculus, either. He did  mention however explicitly infinitely divisible numbers, something  we know now as rational or real numbers. That is all you need to get to  geometric series. This is mentioned by Boyer, and Lynds for example. The  discussion of Zeno's paradoxes could do just with these.
 * Later there were people who used Zeno's paradox to illustrate calculus. That  is very much well sourced as well.  The Mathematics Illuminated  source is an example of such a use. These sources all refer to the  geometric series, which was mentioned in Aristotle's description.
 * That calculus can be used to address mathematical aspects of the paradox is  also well sourced, for example in Boyers book again, or Mathematics  Illuminated source. These also refer to the geometric series,   mentioned in Aristotle's description.
 * What however is not sourced is that Zeno did consider movement at or below Planck length.  And we haven't found a single source for this, yet.  Ansgarf (talk) 07:22, 28 February  2010 (UTC)
 * I am not sure what point of my argument needs more explanation, since it seems that you keep asking the same  question, namely why I think that there is no need to explicitly  consider movement through and below the Planck length when discussing  Zeno's paradox.  Maybe we should give the other some chance to provide  some input. Ansgarf (talk) 07:22,  28 February 2010 (UTC)


 * re "What however is not sourced is that Zeno did consider movement at or below Planck length.  And we haven't found a single source for this, yet" Right then, so we're  now speaking on behalf of Zeno? Right? What exactly did Zeno say? What  records reveal what he said, directly, as written by him?
 * "And we haven't found a single source for this, yet" -- try a wee little  branch of science called ... golly, forgot its name, hang on, getting  there. Uhm, it'll come back to me, oh, yeah, almost forgoet, it's called  'physics'.


 * As before, the onus is upon those making assertions to back them up with credible Reliable Sources who address  the issue of movement of Zeno's runner, arrow and tortoise through all  increments, including those at and below the Planck length. Upon what  basis do you reject consideration of movement through and below the  Planck length? What reliable sources confirm the validity of your  rejection?


 * Until confirmed otherwise, the 'high-accuracy' remains both valid, appropriate, and scientifically  credible.Steaphen (talk)  07:39, 28 February 2010 (UTC)


 * You ask "What records reveal what he said, directly, as written by him?". The best  sources are "Aristotle's Physics[1] and Simplicius's commentary  thereon".Ansgarf (talk) 07:48, 28  February 2010 (UTC)
 * "And we haven't found a single source for this, yet" (...) it's called "physics" . Which book?  Ansgarf (talk) 08:05, 28 February  2010 (UTC)

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)

Newest proposal by Ansgarf
Since the discussion between Steaphen and me on my latest proposal is hard to follow, I just want to repeat what  the proposal is. I'll try to keep it short, and I would like to ask Steaphen kindly to hold back on his comments; I understand his point,  but I am not convinced. Sorry. I am really interested in the input of others, so I urge Steaphen to give them a chance to respond.

My proposal is basically to distinguish clearly when we talk about the  runner and tortoise from the paradox (the map), or when talk about  physics theories of motion of physical bodies (the territory). When we talk about mathematical theorems we do not need to qualify the accuracy,  mathematical theorems are mathematically correct (provided they are  proven). This means for the two sentences in contention
 * ""Zeno's paradoxes were a major problem for ancient and medieval philosophers.  More modern calculus has solved the mathematical aspects of the  paradox".


 * I propose to either
 * Revert to the above version
 * Omit the "highly accurate" from the current version. It would become While modern calculus has solved the mathematical aspects of the  paradox, some philosophers ....
 * Or change it to Many philosophers argue that beyond mathematical solutions the paradoxes  remain unsolved.


 * "While mathematics now has formulas that can be used to calculate with high accuracy where and when the moving  Achilles would overtake and pass the Tortoise,[22][7][23][24] ..."


 * I propose to change it to either
 * While mathematics can be used to calculate, given the above description, where and when the moving  Achilles will overtake the Tortoise of Zeno's paradox  ...
 * or, to make the distinction between map and territory even clearer
 * While mathematics can be used to calculate where and when the moving Achilles  will overtake the Tortoise of Zeno's paradox, which when applied to the  motion of physical bodies gives highly accurate results, ...

Any comments?Ansgarf (talk) 07:44,  28 February 2010 (UTC)


 * The simplest case in which calculus applies is not Achilles and the tortoise, but in the dichotomy - which  involves no measured distances and no measured speeds - in fact no  measurements at all. It clearly is about the map and not the territory  & clearly there is a multitude of support that all mathematical  aspects of this paradox have been solved completely. Usage of "accuracy"  is inappropriate when only the map/model is being considered. Accuracy  comes into play only in agreement of math with reality - and there is no  way to ultimately decide in this case - it is a theory choice (choosing  either continuous space or discrete space model). There is nothing in  quantum mechanics which makes discrete space a better model/theory than  continuous space. Moreover, as I see it, there is a major problem in  thinking objects "jump" through discrete space - conservation of  momentum becomes impossible.
 * Achilles and the tortoise can also be phrased without using any numerical measurements, but the math is more  complicated, the size of the bodies can become an issue, as can the  meanings of "catch up" vs "overtake". The dichotomy need involve only  some unspecified distance (d) - which is not measured, but is  conceptually imagined. The sum of all the infinitesimal components of d  is (surprise?) d - and not with any relative degree of accuracy, but  rather exactly d. That we can never actually measure any distance  with exact and absolute precision does not matter to this conceptual  model.
 * Furthermore, it has never been demonstrated anywhere that "space" is an entity with any properties at all - it has never been  demonstrated that space is anything more than a model we have  "hard-wired" into our brains. The only "property" of space that is even  proposed in modern physics is that it is "curved" - this is an  alternative to using gravity to explain curved paths taken by matter  & by energy in motion. It remains to be seen whether saying space is  curved forces us to say space is some unique kind of ontological  entity, or whether an alternative ontology will work. If we are not sure  if space is a entity with any properties at all, any debate about  whether it is discrete or continuous can only be preparatory to further  investigation - it cannot be resolved with present knowledge. We are  free to use whichever model "works" - and I am not aware of any  situation in which discarding conservation of momentum "works".  --JimWae (talk) 20:51, 28 February  2010 (UTC)


 * In the case of the Dichotomy Paradox movement through increasingly smaller (1/2 sized) increments of distance was  queried by Zeno. As before, the onus is upon those making assertions  that any such movement (through 1/2 the preceding distance) is supported  by 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) 22:58, 28 February 2010 (UTC)

As explained long ago and many times since, the term "accuracy" applies only when  comparing a model to reality. It does not apply within the model.--JimWae (talk) 23:55, 28  February 2010 (UTC)


 * Good. I move that, on the basis of JimWae's statement, we include the field of astrology as a model as well. That  when Uranus flies past Pluto, Achilles catches the tortoise. Since there  is now a clear statement that we need not b rigorous with matching  theory with reality, astrology, numerology and others are models that  can be included.


 * Thanking all, including the astrologers. Steaphen (talk) 00:10, 1 March  2010 (UTC)

The continuous model of space has not been discarded by science. Comparisons with astrology are rhetoric without relevance--JimWae (talk) 07:59, 1  March 2010 (UTC)


 * That is your opinion. What is the evidence that shows your theories are any more relevant at the  Planck scale than Astrology? And can that evidence (e.g. by way of  experiment) be independently verified? Steaphen  (talk) 03:05, 2 March 2010 (UTC)

Which country, which map?
I propose that we seek to find a map (mathematics, or similar) that is appropriate to the territory.

Ansgar, in the above two sections, as far as I can tell, has found a map, and  believes it applies to our territory. But he has provided no Reliable Sources that confirm that his map is relevant to our territory, beyond a  few crude similarities and approximitions.

That's like suggesting a map of Australia, is good enough and close enough to safely  and competently navigate through Austria.

Are we still talking about an exacting science here? Are we still seeking solid, robust resolutions to Zeno's Paradoxes, that will withstand the scrutiny and  test of time?

As before, the onus is upon those making assertions to back them up with credible Reliable Sources who address the issue of  movement of Zeno's runner, arrow and tortoise through all increments,  including those at and below the Planck length. 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) 07:57, 28 February 2010 (UTC)


 * "Are we still seeking solid, robust resolutions to Zeno's Paradoxes, that  will  withstand the scrutiny and test of time?": No. Absolutely not. Not our  job. We report on the literature, just that.  Paradoctor (talk) 22:17, 1  March 2010 (UTC)


 * And which literature would that be? That which agrees with your bias and speculations? Steaphen  (talk) 03:01, 2 March 2010 (UTC)


 * You should be careful who you accuse of bias and speculations. I  have an ego a nuke couldn't hammer down to size, but other users may  tend to feel aggravated.
 * "which literature": All of it, we are an encyclopedia, not a lobby. Paradoctor  (talk) 03:10, 2 March 2010 (UTC)


 * I'm happy to allow others the freedom to speak for themselves. Who did I  accuse, and of what, other than to ask questions? Do you wish to now  censor questions? "All literature" ... "ALL?" including books on  Astrology? Too many questions for you?  Perchance you come from a  culture that in the past squashed the most sacred freedom of life: to  ask questions, to think outside the box, to explore new horizons, to be  an individual? (Just asking).Steaphen (talk) 03:49, 2 March 2010 (UTC)


 * "including books on Astrology?": Can you cite one that passes WP:DUE?
 * In your directly preceding post, sentences 1, 3, 6, 7 are entirely off-topic. The last one constitutes a personal attack, I want  you to remove it. If you don't comply with my request, be aware that  this will have  consequences. Paradoctor (talk) 11:12, 2 March 2010 (UTC)