Talk:Zeno's paradoxes: Difference between revisions

Resolution of bias and POV

Subsequent to the above, a POV has been inserted on the main page, pending resolution through Formal Mediation.Steaphen (talk) 06:34, 28 November 2009 (UTC)

I think that the discussion is not about the content of the Zeno paradoxes, but about there are some people that think the paradoxes are not. Maybe we can set it as "Mainstream physics claim that paradoxes are solved" and that's it. And maybe we have to set a better mainstream explanation of the solution of the paradoxes. I read a lot of inconsistent claims in this discussion page, but it is not really worthy lost time in this, just open the possibility of another interpretation and that's it. —Preceding unsigned comment added by 190.247.72.15 (talk) 23:36, 10 December 2009 (UTC)

The mediation has been called because there are statements on the main page expressing unsupportable points of view. The statement "using ordinary mathematics we may arrive (or calculate") is fundamentally wrong, and needs correcting. If you think statements that have no basis in fact are able to posted in an encyclopaedia as being fact, then your definition of what constitutes an encyclopaedia is worlds apart from mine.Steaphen (talk) 02:51, 14 December 2009 (UTC)

btw, hands up allll those competent physicists (not mathematicians off in cloud-cuckoo land) who believe we can precisely calculate the location and speed of anything. Anyone? Are there any physicists, any at all, willing to commit career suicide by stating categorically we can calculate location and speed of stuff -- irrespective of size -- at and below the Planck length? Please also provide the research institution at which you work used-to work.Steaphen (talk) 03:11, 14 December 2009 (UTC)

I am not quite sure why you want to exclude mathematicians. In the end the arguments count, and your arguments aren't dismissed because of the fact that you are neither a physicist nor a mathematician. They are dismissed because of the content. That you believe that mathematical proofs are less rigorous than physical experiments, and subject to experimental validation, doesn't speak for your credentials in either area, but it wouldn't exclude you from making a valid point.

The problem is that Steaphen (currently) wants to remove any mention of any algebraic description of motion. To clarify this position to 190.247.72.15, lets give an example.

Suppose you have an object A that moves at twice the speed of object B. Suppose further that object A start at 0, and object B at 1. Assume that this speed is 1, then the position of object A can be described by $p_{a}=2t$, and the position of object B would be $p_{b}=1+t$, where t is time.

Steaphen now claims that we cannot solve these equations for t. He say that it is fundamentally wrong to solve them, and wikipedia should nowhere on wikipedia say that such equations can be solved. Steaphen even goes further and claims that you should not even mention the equations in the first place, because they are fundamentally wrong. He even goes further and claims that no physicist in his right mind would use algebraic equations to describe motion. It might sound extreme, but this appears to be his position. Please correct me if I am wrong.

To insert "mainstream" would give undue weight to a fringe position, that as far as I know only Steaphen holds. You will be hard pressed to find a single book on physics that does not use mathematical, geometric or algebraic descriptions of motion. Any textbook, look for example at Kreizig's Advanced Engineering Mathematics, but also any scientific paper, like Bohm's A suggested interpretation of the quantum theory in terms of “hidden variables”, Phys. Rev. 85, 166(I) – 180(II), 1952. As said, you will be hard pressed to find one, pulished in the last 100 years, that doesn't.

If Steaphen complains about unsupported, speculative and demonstrably erroneous suppositions, then it might be him who is sitting in the glasshouse.Ansgarf (talk) 03:50, 14 December 2009 (UTC)

I personally feel that the algebraic solution should not be included because it's unrelated to Xeno's paradox, which is a sigma addition problem, not an algebraic problem. The algebra does not solve the 1/2 1/4 1/8 series. Sigma addition does. --71.213.238.190 (talk) 16:48, 16 December 2009 (UTC)

A runner leaves the starting blocks. Picking his nose as an example location on which to focus, his nose (along with the rest of his body) beginning moving. It (his nose) moves 1/10 of a Planck length (he's quick to finish). What mathematical/geometric/algebraic expression can predict or plot his nose's movement? Explain what experimental data and theories support your thesis. Hey, was that Heisenberg rolling over in his grave (Not to mention Bohr, Bohm, Schrödinger et al)? Oh dear, someone has let the cat out of the bag. It's run away, but wins by a nose. Steaphen (talk) 04:48, 14 December 2009 (UTC)

(Ansgar) You are indeed wrong. Perhaps English is not your first language. Reread my words. Please be more exacting in your analysis. I said, and this is quite clear, no-one is able to state categorically, with any credibility or substance that we may precisely calculate the physical qualities of momentum and location of anything. By all means apply your theories, your mathematical expressions, but there is NO evidence they can be correlated to actual physical reality -- you know, like the actual movement of things like, gee I don't know ... arrows, runners, tortoises. Yes, of course, none of this has any relevance to the issue of Zeno's Paradoxes, the apparent paradox of movement of physical things. Right. Which planet are we on? Cloud-cuckoo land? As for "mainstream" ah, yes, the crowd opinion. Do I smell smoke? As I said, the nonsense on this page beggars belief. Where's that mediator? Steaphen (talk) 04:12, 14 December 2009 (UTC)

Steapen, according to your own demands you just made a fundamental error, because under your own definition it is wrong to state that a runner is at a 1/10 of a Plank length distance from anything. And you gave yourself the algebraic equation, even if you phrased it in natural language. Namely that the movement is 1/10th of a Planck length. And Heisenberg's Uncertainty Principle is still just about the measurement of momentum and position.

You probably know that the level of Planck distances, the point of the nose is an abstraction at best. And you probably know that at Planck distances the QM description of the particle that forms the tip of the nose is given by the wave function, which can be interpreted as a probability distribution. And according to Ehrenfest's theorem the centre of particle that is the point of the nose behaves like a classic particle, and its behaviour can be described by an ordinary differential equation. And if this is too deterministic for you, use Schroedinger's equation, it also describes the motion of the particle that forms the point of the the nose, mathematically, but in a bit more detail.

I am surprised that you want me to be more exacting, since I got the impression that you found it already fairly burdensome. But maybe you are right, and I have been to easy on you. Anyway, I am not quite sure why you think that I misunderstood you, if you are actually confirming in the same paragraph my interpretation. You just said "By all means apply your theories, your mathematical expressions, but there is NO evidence they can be correlated to actual physical reality", didn't you? And this while you would be hard pressed to find a single physicist who does not believe that their equations are correlated to actual physical evidence. This is entailed in the very definition of being a physicist; using mathematical tools to describe physical phenomena, and then try to find experimental evidence. That is the reason why I actually object to include the qualifier "mainstream", since there would be no actual physicist who shares your view that you cannot use mathematical, geometric or algebraic means to describe motion. Ansgarf (talk) 12:24, 14 December 2009 (UTC)

re your "because under your own definition it is wrong to state that a runner is at a 1/10 of a Plank length distance from anything." no, that is not my definition, it is my question. What happens at those scales.

re your "And Heisenberg's Uncertainty Principle is still just about the measurement of momentum and position." WRONG. it is about the relationship between momentum and position. It is independent of all measurement. The Uncertainty Principle is a PRINCIPLE. Again, it must be that your first language is not English. If you tell me what it is, I'll see about speaking in your native tongue. I might have some trouble with Swahili though.

"I am surprised" ... I would think you're mostly surprised by everything.

I loved this "the point of the nose is an abstraction at best." Priceless. Absolutely priceless. You must get somebody else to pick your nose for you. By your definition, you can't pick it. Did I say, 'priceless.'? It must look awful to repeat myself so much.

And this (it only gets better. Hell, this is better than any entertainment you'd pay for): "And you probably know that at Planck distances the QM description of the particle that forms the tip of the nose is given by the wave function, which can be interpreted as a probability distribution." A probability distribution? Wow, that's really impressed me. So your nose, that you can't pick, is probably there, exactly where you calculate it to be with your slide rules and equations? So, the mathematics is precise about the possibilities and probabilities, but not the actual particles.. Gee, I wonder what that says. How many priceless moments are you allowed?

"the point of the nose behaves like a classic particle," you're not serious, are you? There's that question I need to keep asking "he's not serious, is he?". "we can calculate" requires ABSOLUTE precision and determinism. No if, or buts, or 'acts like" ... "acts like"? You're not really serious, are you, you're just teasing me aren't you?

But waaaaiit, there's more: "hard pressed to find a single physicist who does not believe that their equations are correlated to actual physical evidence." Right. Name one who will argue that we can correlate their equations (precisely) with the actual physical evidence (even theoretically, for distances around the Planck scale)? Just one, name one itty-bitty little short physicist, maybe, or one lying down who's still asleep, or even a dead one. Hell, I'm not choosy.

Priceless.

Steaphen (talk) 22:06, 14 December 2009 (UTC)

You know, I'm not sure why, but whenever I read the replies on this site by Ansgarf et al, I'm reminded of Monty Python's "Life of Brian". So, Ansgar, you'd like to have a baby (idea). But where's it going to gestate? In a box? Yes, you're all individuals. Well, I'm not. What has Quantum Theory ever done for us? Uhm, most success. Well okay, beside most success in predicting reality, what else? Er, ah, enabled DVD players, and lasers and a whole stack of really cool things. Right, what else? Brought new insights into possibilities? (paraphrasing) REG: Oh. Possibilities? Shut up!

Steaphen (talk) 22:26, 14 December 2009 (UTC)

Can you please make up your mind. Do you want me to be more exacting, or to just to "shut up"? I'll try to be more burdensome first, if that is o.k with you.

• When you say, what happens if a runner is at 1/10th of a Planck length you are assuming that he can be at 1/10th of a Planck length distance. So, can you make up your mind whether distances can be smaller than a Planck length or not?
• The article on Uncertainty Principle says In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. The website of American institute of Physics says about the Uncertainty relations The uncertainty relations have to do with the measurement of these four properties; in particular, they have to do with the precision with which these properties can be measured.[1]
• No, you might not be able to pick your nose exactly, and you mentioned yourself repeatedly that the exact position of anything can not be determined exactly. So, could you make up your mind as to whether a point of the nose is a deterministic point mass, or not. Because in your question you assume it is, in your answers you ridicule it. So what do you want to assume?
• So, you do ridicule the notion of distributions, while you at the same time embrace QM? You might know that QM, the most successful theory, uses distributions over possible states at its core. So, please could you make up your mind, do you think that the distributions in QM provide a successful theory that has been experimentally validated, or do you think its description based on distributions is utter nonsense that has no correspondence with reality?
• I am not teasing you when I point to Ehrenfest's theorem, although, to be frank, I do enjoy to see how you struggle with the concept. When you assume that an object is defined by a wave-function as QM does, you can define its centre as a point. There is nothing particularly impossible about this. You can define the centre of an object, even if it is not a point itself. And if you do this for the wave function, you end up with an ordinary differential equation. That is the core of the theorem. Or do you see any mistake in the theorem?
• When a physicist writes down an equation that describes behaviour, he really says that the object in question behaves like that equation. Although, he'll probably admit that you cannot measure it precisely. I get the impression that you are not too familiar with the scientific method. It is not required to have have confirmed every theoretically possible prediction to accept a theory. To the contrary, a theory, if consistent, can be accepted unless it is falsified by previous or current experiments. These experiments confirm only all actual predictions, not every possible prediction. Your problem actually is that at Planck scale it might be very difficult if not impossible to falsify any prediction. How do you check that your runner is not already past his finish?
• I enjoy your references to popular culture. It is actually ironic that you tell me to shut up about distributions in QM. There is a famous quote, frequently attributed to Feynman, but that was actually coined by David Mermin, as response to people asking what really happens below quantum level. The quote is "Shut up and calculate!". It seems that at least this physicist thinks that you can calculate. Ansgarf (talk) 01:14, 15 December 2009 (UTC)

Dear Ansgar, I am so sorry, but I think I've used up my quota of 'priceless' responses -- otherwise I would provide you a detailed reply (I confess, mostly multiples of 'priceless', or variations therefore) but alas, as I said, I've exhausted my stock of 'piceless'es. Besides I start to look a bit silly, repeating myself ad infinitum, like one of your runners on his way, running through his infinite points as he calculates his way through timelessness. Awh, maybe one more. Priceless. Blessings on your path.Steaphen (talk) 04:28, 15 December 2009 (UTC)

Hi Steaphen, we have been over this a few time before, haven't we. And it does not surprise me that you cop out as soon as you are asked to give an exact answer. Neither does it surprise me that instead you try to get away with a few off-topic facetious remarks and references to popular culture. Not that I don't enjoy them. In this spirit I just want to share that your replies remind me of another gem of British comedy. You make very few responses that couldn't be summarised by ""Yeah but no but yeah but no but yeah but...", or "Don't go giving me evils!", or "Shut up! I ain't even dun nuffin' or nuffin'!" and of course the priceless "Oh my god! I soooooo can't believe you just said that!". Ok, enough silliness, let us indeed wait for the mediator to come to a conclusion.Ansgarf (talk) 04:48, 15 December 2009 (UTC)

Steaphan's comments included in this series of posts indicates a complete lack of awareness regarding the difference between simple algebra & calculus. The point about "Using simple mathematics we can calculate..." is to point out that calculus is NOT needed to determine the point at which Achilles catches the tortoise. If we are given each runner's speed and the amount of the head start, then using simple math (6th grade level or less) we can determine the relative position of each runner at every second. With the numbers in the article, Achilles "catches" the tortoise sometime between whole number values for the seconds. Algebra (9th grade math or so) can be used to calculate the specific (fractional) time & distance at which Achilles catches the tortoise. To say it is a "specific" time and distance is not the same as saying we can determine the time & distance to an infinite degree of precision. Each runner's speed is already a rounded-off value, as is the head-start. Most people understand that speeds, distances, and times are not usually given to an infinite degree of precision. I have a proposed solution to this impasse, but I wish first to determine whether settling this one point will settle the controversy, and if Steaphan will be content if the solution does not result in including his beliefdoctor thesis in the article--JimWae (talk) 05:52, 11 December 2009 (UTC)

I must also repeat: The paradoxes are not about whether or not we can find a precise point in space and time at which Achilles is located. (In fact, the arrow paradox depends on the arrow having a precise location at a precise point in time.) Zeno paradoxes do not stand or fall based upon whether we can model motion mathematically to calculate some points along the way. Zeno's paradoxes are based on the impossibility of completing an infinite number of tasks.--JimWae (talk) 06:05, 11 December 2009 (UTC)

I would never suggest that "the fact that measurements are approximate suggests QM is irrelevant". My point is that the uncertainty within measurements made regarding race-courses has a far more significant bearing than QM on how precise our calculations can be - and that it would be ludicrous to introduce QM as the main factor of uncertainty in such calculations. --JimWae (talk) 06:13, 11 December 2009 (UTC)

JimWae, you can't be serious. Regarding your "The paradoxes are not about whether or not we can find a precise point in space and time at which Achilles is located." On the front page you state that "Using ordinary mathematics we can arrive at a specific time when and place where ...". Are you serious? that you can make both statements, and remain credible? Are you aware of the disconnect that your theories require, not to mention that they lack even a modicum of consistency. Enough of this nonsense. Let the mediator(s) sort it, and failing that, the arbitrators. And failing that Jimbo. After around a century of having quantum theory (beginning with Einstein's 1905 paper on the photo-electric effect), there is simply no excuse for clinging to old, deterministic, clockwork-universe beliefs. Steaphen (talk) 04:01, 14 December 2009 (UTC)

There are competent physicists who affirm that we cannot precisely "calculate" the location of anything, no matter what its size or circumstance. Even the moon is accepted as having a wave-function which requires that we cannot exactly calculate its location and momentum (speed) (at and below Planck scale increments). The argument that we can "arrive at a specific time" is not even 'bad science' or any form of science. Such statements fail to observe the basic principle of the scientific-method.

The comments by various respondents on this site have degenerated into the unimaginably absurd. Case in point: JimWae said that

"The uncertainty in finding the point at which Achilles catches the tortoise comes more than 2 dozen magnitudes of distance before QM enters the picture. QM is an issue at around 10⁻³⁵ metres. We measure distances on a racecourse to a certainty of - at the very best - about 5 millimetres. Stopwatches at races record differences only as small as 1/100 second, whereas QM enters as an issue at about 10⁻⁴⁴ seconds. We do not have instruments than get anywhere near the theoretical limits of QM, and introducing QM as the primary uncertainty in a race is absurd."

If we analyse JimWae's comments, he suggests that to solve movement of racehorses, for example, we may theoretically use mathematics/algebra/calculus/geometry to plot their exact location and speed, irrespective of whatever distances they move, including at and below the Planck length. According to JimWae, we can only measure to around 5mm, therefore QM is irrelevant to the issue of Zeno's Paradoxes? He appears to misunderstand QM, in that reference to the requirement for actual measurement is not dictated by the Quantum Theory. The mathematical expressions stand independent of experimental evidence for them. It just so happens that every experiment (and there have been many tens of thousands of them) have not once disproved Quantum Theory. Hence why many physicists readily accept it to be the most successful physical theory, in history.

JimWae and Ansgar both appear to think that it is valid to apply geometric/algebraic/mathematical expressions to some phenomenon, even when there is overwhelming evidence revealing that it is invalid to do so. The Uncertainty Principle disallows application of any mathematical/algebraic/geometric expression to precisely define (to infinite precision) momentum and position of physical matter. The Uncertainty Principle requires that we may only approximate the location and speed of objects, irrespective of whatever mathematical/algebraic/geometric expression is used. This has nothing to do with the failure of measurement. It is simply reflecting the deeper non-deterministic nature of reality. If there are any Reliable Sources who can assert otherwise, they're welcome to state as much.

The statement "using ordinary mathematics we may calculate" requires perfect determinism (requiring INFINITE precision through INFINITE orders of magnitude below the Planck length), and that any such calculation is reflected in fact. The statements in contention and under scrutiny in this mediation have not been of the kind "Using ordinary mathematics we may approximate ...". Instead they have stated categorically, "we may calculate", with perfect determinism, requiring absolute correspondence with reality: no such absolute correspondence has been observed in actuality. Furthermore, the Uncertainty Principle requires that we may not even do so theoretically, by any mathematical/geometric/algebraic means whatsoever. The Quantum Theory and the experimental FACTS work together to reveal infinite-series/ordinary mathematics cannot precisely calculate an objects position and momentum, time and energy. The statement "using ordinary or simple mathematics we can calculate" is so fundamentally and deeply in error, as to be on par with "using ordinary mathematics we may calculate the number of angels on pinheads" ... both statements show an equal lack of correspondence with observable reality, and are therefore about as useful, meaningful or rational.

At what point does Wikipedia rein in unsupported, speculative and demonstrably erroneous suppositions of editors? Steaphen (talk) 04:01, 14 December 2009 (UTC)

Arbitrary break

• I doubt any mediator will want to take this case on unless the discussion becomes more focussed. I see there is no response yet to my statement that I have in mind a possible way to resolve this "impasse" --JimWae (talk) 22:20, 14 December 2009 (UTC)

Impasse? I've simply and repeatedly asked that you provide a Reliable Source who states that we may precisely calculate the momentum and position of a runner, or hare, or any part thereof, at all increments in movement, including at and below the Planck length. It doesn't get much simpler or more focused, does it?Steaphen (talk) 22:31, 14 December 2009 (UTC)

• The text does not say "precisely". Please try to focus. Nor does it say "momentum and position", nor "momentum" at all.--JimWae (talk) 23:46, 14 December 2009 (UTC)

Well, it's either precise or approximate. State your case. Precise (to infinite precision, as required by infinite-series solutions, and all mathematical solutions) or approximate.

Approximate or infinitely precise? If precise, and you define position then you won't have a clue as to the arrow's, or hare's velocity.

Is that clear enough for you? And if "precise" then tell me, what happens precisely at and below the Planck length?

Is that focused enough for you? Precise or approximate? WHICH IS IT? Steaphen (talk) 04:19, 15 December 2009 (UTC)

Dear JimWae, you appear to have sufficient intellectual horsepower to see where all this is headed. Now, to make it easy for you: I believe physical stuff and physical reality exists (at and below the Planck level) in superpositions of possibilities, all of which emerge from, and ride deeper nonlocal fields of potential. In which case, you're peeing into a hurricane if you think you can precisely define bits of "physical stuff" that aren't even technical real, or tangible. So, the best you'll do is "using ordinary mathematics we can approximate ..." yadda yadda. But to argue that you can precisely calculate is, as explained, 'peeing into a hurricane" ... If you want to accept that change on the main page (from "we can arrive" to "we can approximately arrive" or words similar), then we're done, mostly. There's a few other statements that need sorting, but in no way are you, or anyone else, justified in saying "using ordinary mathematics we can arrive (or calculate)." It's just bad science, or not even science at all to suggest theories that are demonstrably wrong. Steaphen (talk) 05:14, 15 December 2009 (UTC)

This discussion could go on indefinitelly. Steaphen you need to accept the fact that there are many different physical models of reality (Newtonian, Einsteinian etc) all of them based in the same language - mathematics. Then, you have the calculus which allows us to calculate (pretty much) anything within your model to an arbitrary precision/accuracy (which only depends on the calculus method that was used). So to say that you cannot precisely calculate a property within a specific physical model, when mathematical formulas are given, is simply false. Within the Newtonian model, I can do exactly what is said: use the ordinary math to calculate all those properties listed, even though the model, and the properties themselves (who is to say that physics in 1000 years is still going to use properties like velocity, momentum?) are, if you want, entirely fictional their only relation with actual reality is that mathematical models behaving similar to our perception of reality, may be found. They are similar, but still fictional. Even QM might be. As for the paradox itself could it be possible that what Zeno really meant was not, 'motion is impossible' but 'motion as we perceive it intuitively, is impossible, therefore our intuitive grasp of motion must be false'. Since motion is clearly possible, his purpose couldn't have been to prove that motion is impossible, so it must have been something else. The paradox itself rests on the continuous (or, at minimum, dense) model of space and time, therefore, maybe what Zeno actually wanted to imply is that our intuitive, continuous model must be false, and that reality works in a different way from what we perceive. A number of questions stem from that speculation, for example why would the mechanism for interpreting our reality be false, it sounds like something that nature doesn't usually do. Sort of like giving us hands that we couldn't use to pick up things with... On the other hand this does seem to validate some more discussion on the QM topic within the scope of the article. As others have pointed out, this is an article about paradox so references to physics should be kept to required minimum, still the primary topic of this paradox appears to be motion, so to me it makes sense to include as much essential information humanity has gained about motion so far, as possible. Cheers, Zibbo. (89.142.158.223 (talk) 09:24, 15 December 2009 (UTC))

JimWae, Ignoring the somewhat nonsensical responses by others: precise or approximate?

State your case. If precise, then as above, if not, then what justification can you make for 'we can calculate'?

Cheers, Steaphen (talk) 19:19, 15 December 2009 (UTC)

• The precision of the calculation is limited only by the precision of the measurements of distance and speed, just like all calculations using measurements are. When measurements are used, there is no absolute precision - all precision is relative. Calculations using measurements are not themselves approximations, the measurements are what is approximate. The calculation produces a quantity, say time (in seconds), the precision of which depends on the precision (the significant figures) given in the measurements. Neither Achilles nor the tortoise can run at a constant speed over the entire race - each must accelerate to start. 11 1/9 seconds is more specific and more precise than "somewhere between 11 seconds and 12 seconds"--JimWae (talk) 23:17, 15 December 2009 (UTC) ---- The implication of the Planck units is that we will never have instruments able to measure quantities smaller than them. (We are not even close with our present instruments.) We cannot know for certain what happens between Plank lengths & Planck times, but by continuing to use a "continuous model" at that level, we do not have to discard laws of physics such as the conservation of momentum. We do not have to conclude that space and time are some kinds of entities with a "fabric" composed of jumps, just like we do not have to conclude from looking at still frames from a movie that the subject actually "jumped" in space.--JimWae (talk) 02:03, 16 December 2009 (UTC)

Good, I'm glad we're in agreement. Due to angles on pinheads, that we can't actually see or verify with our instruments, we may conclude that Zeno's Paradoxes are solved by said angels transporting runners and hares and the like. No evidence, buy hey, it's only because the instruments can't see them.

You have now stated that the quantum theory, as in the wave nature of matter, is, at a root level, still able to be precisely determined. Hands up all those physicists who agree with JimWae that we can precisely calculate the physical characteristics of physical things. Anyone?

While it is entertaining to watch the contortions to which people go to defend the indefensible, nonetheless, it behoves all of us that this nonsense is stopped.

Unless you can provide a reliable source stating that we can precisely CALCULATE the whereabouts (speed and location) of physical things like runners, and hares (as in the quandry first proposed by Zeno) -- including and especially at and below the Planck length, I'll update the front page to say "approximate" where required. You have posted POV, with no supporting Reliable Sources. Steaphen (talk) 06:19, 16 December 2009 (UTC)

• Which of every high school algebra, high school physics, and college physics textbook is not a reliable source? Are you not at all interested in a proposed way out of this impasse? I repeat, the text does not say we can "precisely calculate". Precision is a relative term (as is approximation) -- neither is a useful adjective in this context and both would be POV. --JimWae (talk) 08:19, 16 December 2009 (UTC)

• I see you've gone ahead and started what could well become an edit-war. It is pure syntactic error to write "approximately arrive at a specific time" and "approximately calculate" --JimWae (talk) 08:30, 16 December 2009 (UTC)

Once again, there is no justification for saying "we may arrive" -- it implies we may accurately do so. The school texts don't cover the issues we are dealing with here ... namely, movement through the Planck scale.

If you cannot furnish a reliable source, you are pushing a POV. However, you are not justified in asserting that my "approximate" is a POV, in that you have not established a case for 'precise' which is implicit in the statement, "we can arrive". In other words, the onus is not upon me to 'prove' approximate' it is upon you to 'prove' precision, which is implicit in the statement "we may arrive'.

Furthermore, if you don't believe "precise" is implicit in saying "we can arrive" than you will not argue with clarification "we may approximately arrive".

In any event, Zeno's Paradoxes is about the precise means by which movement occurs (it is about the precise means by which Achilles catches the tortoise). That's why they have caused serious thinkers difficulty for 2,400+ years. In that context, precision is an absolute, unremitting requirement for any valid treatment.

You have not provided precise explanations, and therefore the statements "we may arrive" need to be prefaced with 'approximately' etc.

It would seem no mediators are going to step in. Arbitration will be called in due course. Steaphen (talk) 09:55, 16 December 2009 (UTC)

I see that you changed my addition of 'approximately' to the article. I'll expedite the call for arbitration.Steaphen (talk) 10:04, 16 December 2009 (UTC)

• "approximately arrive at a specific time" is semantic and syntactic gibberish --JimWae (talk) 10:40, 16 December 2009 (UTC)

that's a lame comment. You knew it was a simple grammatical error. It should have read "approximately arrive at a time" . Seriously, I expected better of you.Steaphen (talk) 12:43, 16 December 2009 (UTC)

• Apparently you revert before you read comments here (before you can "decide" if they are "lame" or not), otherwise you would not have reverted to the syntactic goop that you wrote. By the way, I have no problem in saying that the value calculated has limited precision, but it is not the calculation itself that is approximate, it is the resulting quantity of the calculation that has limited precision (otherwise we have goop again). But the level of precision with the numbers given in the example is already far less than the level of precision at which QM would be a paramount consideration. It's all (QM included) about the imprecision of measurements.--JimWae (talk) 22:58, 16 December 2009 (UTC)

You may be proficient in mathematics, but your understanding of quantum theory is simply wrong. The quantum theory does not allow precise knowledge (e.g. of location and speed, time and energy) of physical matter (including that which comprises arrows and the like). This is independent of the actual measurement. At the root level, matter exists in superpositions, which 'coagulate' in our reality as a point particle, a point arrow, etc., but the indeterminacy remains at the root level. that is why the position and speed of an arrow cannot be precisely calculated, because in real terms, it isn't even there, until we observe it.

It seems I'll need to write an article for the arbitrators, listing all those physicists and their quotes concerning the root level indeterminacy of physical reality, and the inability to say, with any substance "using ordinary mathematics we may arrive" or "calculate".

If you want to leave in "we may calculate" for Zeno's arrow, fine, but include angels on pinheads as well, because each have been as equally substantiated in fact (i.e. none).Steaphen (talk) 06:02, 17 December 2009 (UTC)

JimWae, simple question. Do you believe that at the root level of physical reality, at and below the Planck length and time, we may IN THEORY, precisely determine location and speed of physical stuff, at every and any point we choose? Leave aside any reference to measurement. Take it out of the picture completely. IN THEORY are we able to precisely calculate speed and location of physical particles? Simple question, yes or no. No need to waffle on, or deflect or avoid the issue, just a simple answer "yes I believe we can precisely calculate the precise location and momentum of physical matter at and below the Planck length and time" or "no, I don't". NB - this question has been repeated below, with additional commentary.Steaphen (talk) 22:10, 17 December 2009 (UTC)

Revert

It is extremely rude to make changes to an article that are currently under mediation. I'll revert the paragraphs affected by Steaphen's recent edits, pending mediation, back to the what they were on 7 December.Ansgarf (talk) 11:40, 16 December 2009 (UTC)

While I agree that the paragraph on computing when Achilles can pass the tortoise can be improved, or alternatively could be even omitted completely, as long as it is tagged POV, and we are waiting for mediation, we probably should keep it as is. Ansgarf (talk) 21:47, 17 December 2009 (UTC)

Righteo then

All of you stop writing to the article, kay? There's a difference between a mathmatical model, a philosophical model, and reality. We accept this when we attempt moddling. It's theoretically possible to determine the position of the runner to within plank's measure (Not constant, that's something different but related to plank's measure... which is related to plank time and plank space) distance of a single point, (Actualy it's not but that's because you're determine it's relation to a point that you don't actualy know where it is and it gets really messy but it can be simplified to say "You can to no better accuracy (Always worse accuracy) then plank's constant) and we can do this with a fairly simple experiment. But that's entirely irrelevant. Completely and totaly irrelevant.

Similarly the fact that we can calculate the theoretical position with algebra is irrelevant. All of this is irrelevent.

All we need is a rundown of the paradox, a rundown of the three (Two rather) solutions, and a run down of why philosophers disagree.

If the article is only about the historical problem, by all means, delete all that is superfluous to the 'rundown of the paradox'. Interesting pun.

I.e. delete the section "proposed solutions" -- scrap them entirely, since none are substantiated in fact, and are all POV. Or include the angels on pinheads as well. Equally valid, and equally substantiated.Steaphen (talk) 22:04, 16 December 2009 (UTC)

Needs rescraping

This page has gotten too complex again, and needs to be started clean... again. *Sigh* 71.213.238.190 (talk) 16:27, 16 December 2009 (UTC)

If you find this page tiring "sigh", I suggest you go lie down for awhile, and leave it to those with focus, energy and ability to sort out the nonsense.Steaphen (talk) 22:08, 16 December 2009 (UTC)

Bad math

"Another proposed solution 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."

To whomever said that... Euclidian space (Which is what we live in) is hausdorf, regular, normal, metric, locally compact, and Lindiloff. That means that, amoung other things, between any two points there is another point. --71.213.238.190 (talk) 16:44, 16 December 2009 (UTC)

"Euclidian space (which is what we live in)... is hausdorf, regular..." Really? We live in Euclidean space? Really? What evidence do you have for that supposition, or that the reality we live in, is "hausdorf, regular ..."? Have you read and understood anything on this page? Upon what basis do you make that assumption? What experimental evidence supports your wild assumption.Steaphen (talk) 22:13, 16 December 2009 (UTC) btw, it's bad English to say 'math', which is an abbreviation of mathematics.

71.213.238.190 you fell right in the middle. While I don't agree with Steaphen on this page that often, this topic has been discussed on this page exhaustively. First, space-time uses Non-Euclidean geometry, but this is just an aside, and not relevant for the paradox. But whether it is Haussdorff is relevant; or whether space is dense to be precise. There are many people, like Steaphen, who assert that space-time is not dense, but discrete. Which might be wrong. But even if true, if space-time is discrete, then you wouldn't be able to construct an infinite series of tasks. The statement you mention is not bad math, but part of a case distinction. The case it describes might be bad physics, but isn't bad math. Ansgarf (talk) 22:51, 16 December 2009 (UTC)

The point of this mediation (and it seems, arbitration) is the statement "using ordinary mathematics we may arrive" , or 'calculate'.

Discrete space or dense space is irrelevant to this mediation. The statement "using ordinary mathematics we can calculate (or arrive)", is, on the evidence, plain and simply WRONG.

As before, Zeno's Paradoxes is about the precise means by which movement occurs (it is about the precise means by which Achilles catches the tortoise). That's why they have caused serious thinkers difficulty for 2,400+ years. In that context, precision is an absolute, unremitting requirement for any valid treatment.

To suggest we need not concern ourselves with whether the calculation is precise or approximate, and yet assert that this is the precise explanation for how movement occurs, is a contradiction that beggars belief.

either the calculation perfectly reflects the actual location of an arrow (or its lead atom), or it's approximate. If it perfectly defines its location, then how is it that this process (of using 'ordinary mathematics') clearly, unequivocally and repeatedly has been shown to be wrong -- it doesn't work. It doesn't fit the facts (of being able to perfectly predict/calculate the location of physical stuff at small increments in movement).

Any competent physicist reading this page would agree that we can't precisely 'calculate' the position and speed of anything, no matter what its size. To then suggest we can precisely account for physical movement while ignoring this key fact is just plain ... well, I'd have to say, stupid.

In any event, does anyone reading this page, really, genuinely believe that when we blink an eye, or lift a finger, we move the finger or eye-lid through endless, endless, endless, endless, endless, endless (endlessly repeated) little physical movements? Does anyone seriously believe we physically do that?

Is there anyone reading this page who has the courage to not hide behind descriptions and mathematics, who says 'yes, I believe I move my hand through an endless sequence of little physical movements"?

If there is someone so courageous, explain to me, like a 5 year old, how do you physically do that? Move through an endless number of 'infinitesimal' real little steps? And in each of those steps, minutely and infinitesimally small, below the Planch length, what's happening in that space, because physicists sure as hell have no idea.

Why don't you enlighten them, and save them the expense of operating the LCH

Steaphen (talk) 06:27, 17 December 2009 (UTC)

You should know by now that I am not joking. But I am used to the fact that whenever you can't admit that you are wrong you feel the need to flamebait. Also, I noticed that you are repeating yourself, but not just the arguments, but almost verbatim. I'll skip some of your old arguments, and just respond to some that are somewhat new.

First, if you think that the difference between a discrete model of space and a dense model doesn't matter, explain, how you define an infinite series of non-zero distances, on a finite set of points in space. You don't need to give a lengthy argument, just give the series.

Furthermore, I didn't say that we need not to be concerned about the accuracy of the computation. Accuracy is an important issue in numerical analysis , but that has little to do with QM. I said that applying QM to mathematics is a category mistake. I coincidentally chatted with a researcher from the LHC recently, and he called questioning accuracy of mathematics for quantum systems a red herring.

To illustrate why it is besides the point take the following set of algebraic equations $p_{a}=2t$ and $p_{b}=1+t$. For which $t$ will $p_{a}=p_{b}$? The solution is $1$. Not approximately $1$, because there simply does not exist a value other than 1 that could be a solution. You mention angles on pinheads frequently, implying that you can make up solutions for mathematical descriptions at will. I wonder, can you think of a value for t, that is not 1, and that would solve the equation?

Whether we move though an infinite or finite number of steps whenever we make a movement is in my humble opinion still an open question. A finite number of steps might make some things easier to explain, but I don't see much of a problem either way. But I commend you for making a comment that is actually related to Zeno's paradox. Because, the uncertainty in QM isn't.

Finally a comment unrelated to your latest reply. On Friday you ridiculed the use of distributions [2] to describe physical objects, but since then you have shared with us already twice that you believe that matter exists in superpositions. The way you argue seems to be the following: First, you ask people what to assume that an object is at a certain point in the order of a Planck length. Then you have the following strategy:

• If that person uses your assumption that the object is at the point you ridicule them as ignorant because at that order of magnitude particles are best described by superpositions and not points.
• If that person points out that that at that level particles are best described by superpositions, you ridicule them because they cannot even pick a point.

Of course, maybe you use the word "superposition" unaware of its meaning. Or what reason do you have to criticise me for referring to distributions, and Jim for not referring to them? Ansgarf (talk) 08:59, 17 December 2009 (UTC)

Note: This above reply is based on thisversion. Just another observation. You do not only have split opinions on distributions, apparently. In this thread you first pound 71.213.238.190 for suggesting that space/time is not discrete but Hausdorff, and just one reply later you claim that you couldn't be bothered at all whether spacetime is discrete or not. The only constant here is that your replies look like flamebaits. Although I have to grant you that you toned down your last reply a bit. Ansgarf (talk) 09:18, 17 December 2009 (UTC)

As before, blessings on your journey. Steaphen (talk) 21:47, 17 December 2009 (UTC)

JimWae, (in case you missed this question above): simple question. Do you believe that at the root level of physical reality, at and below the Planck length and time, we may IN THEORY, precisely determine location and speed of physical stuff, at every and any point we choose? Leave aside any reference to measurement. Take it out of the picture completely. IN THEORY are we able to precisely calculate speed and location of physical particles? Simple question, yes or no. No need to waffle on, or deflect or avoid the issue, just a simple answer "yes I believe we can precisely calculate the precise location and momentum of physical matter at and below the Planck length and time" or "no, I don't".

Can you reference any competent physicists who agree that we can IN THEORY precisely calculate position and momentum of physical stuff at and below the Planck length and time?

Any at all? Steaphen (talk) 22:14, 17 December 2009 (UTC)

• It would be rather ridiculous to leave measurement out of considerations of whether we can determine the speed and location of things. We cannot locate anything at all -- not even approximately --- without some framework of measurements.--JimWae (talk) 22:55, 17 December 2009 (UTC)
• Btw, please see this edit. I think you will find less to object to in it.--JimWae (talk) 22:59, 17 December 2009 (UTC)

You have avoided the question. Consider it a thought experiment, in that say in 500 years time they invent some amazing new device or some such that does what we can't now. Whatever. Is it possible IN THEORY, to ever (as in EVER, say in one million years) to precisely calculate position and momentum of stuff?

Yes or no? Simple question. It's a rhetorical question, because your statement "using ordinary mathematics we can calculate" requires that in theory you may do exactly that. In any event, your refusal to answer this question, despite asserting that it is possible via "using ordinary mathematics we may calculate/arrive" confirms the affirmative that you believe it is possible in theory. Now find a competent physicist who agrees with you.Steaphen (talk) 23:46, 17 December 2009 (UTC)

• I cannot think, nor do I think anyone can, of how to locate anything without using measurements. So, the question answers itself - we cannot (& will not ever) measure the exact location of anything to a degree of precision below Planck levels, so we cannot (& will not ever) determine such a precise location. That does not 'mean that space has jumps.
• And neither the paragraph you object to, nor the one I recently put up (which I think is tighter & leads better from one sentence to the next), requires what you say it requires.
• Besides, if we can show mathematically that Achilles can actually pass the tortoise, then we have gone at least part of the way to casting doubt on Zeno's arguments that Achilles can never catch the tortoise --JimWae (talk) 00:03, 18 December 2009 (UTC)

JimWae, you have again avoided the question. You either believe it can be CALCULATED in theory, or it can't. By your statements you believe position and momentum can be calculated. That is your statement, "using ordinary mathematics we can calculate". Your own statements confirm you can calculate position and momentum.

No competent physicist will agree with your assertion. None. Find one that does and I'll show you a physicist without (or soon to be without) a career in physics.

Your statements "using ordinary mathematics (or algebra, or whatever, by any means) we may calculate (at and below the Planck length) ..." is wrong. No physicist will agree with you. None. The front page is simply wrong. Find one physicist that confirms your POV. Just one. Steaphen (talk) 00:47, 18 December 2009 (UTC)

• It is ridiculous to keep repeating the question. I have answered it fully already. Location (for one) cannot ever be determined to a degree of precision below Planck units - BECAUSE we cannot ever measure below that level. It does not follow from that that space has jumps.--JimWae (talk) 01:00, 18 December 2009 (UTC)
• Nowhere in the article is there any suggestion that calculation to such a degree is possible. We do not measure to such a degree when determining if one runner has caught up to another. If Achilles can pass the tortoise, he has more than caught up to him.--JimWae (talk) 01:20, 18 December 2009 (UTC)

You have, yet again, avoided a very simple question: Do you believe the particle, or arrow or whatever, is physically 'there' for any calculation to be performed, irrespective of whether it can ever be experimentally verified. I'll answer for you. You believe physical stuff is entirely still physical, tangible and real, at every level and point down to infinitely short length and time. That is your belief, simply reflected by "we may calculate" .. otherwise, what is it that you are calculating if it is not something physical?

As before, you will not find any competent physicist agreeing with you. The front page is your POV, and unsupported by any competent physicist (Reliable Source).

Steaphen (talk) 01:29, 18 December 2009 (UTC)

• Are you trying to read my mind, now? You are misrepresenting my thoughts, my posts here, and what appears in the article - such has no applicability to what the article should cover. Just because we cannot assign an infinitely precise number to the location of an object, it does NOT follow that space has jumps. I have answered fully, and I will not repeat myself again.--JimWae (talk) 01:37, 18 December 2009 (UTC)

With all due respect, it is you who have stated "we may calculate" (to infinite orders of magnitude below the Planck length). That is a requirement of your statement since you have not stated limits to that calculation. "We may calculate" implicitly covers all orders of magnitude below the Planck length. ALL. Including distances such as 10^{-1,000,000,000,000,000,000,000,000}. and onwards to 10^-infinity metres. No competent physicist on this planet, or any other is going to agree to that. The front page is wrong. Plain and simple. Steaphen (talk) 01:46, 18 December 2009 (UTC)

btw, what on earth has "space has jumps" or not got to do with your statement "using ordinary mathematics we may calculate"? You've now mentioned it twice, yet it is completely irrelevant to the very simple question I have asked you: can you calculate the position and momentum of physical stuff all the way down to the infinitely short (distance and time). You require that you can. Fine. Find one physicist who agrees with you.

let's be clear on this. The mediation was called because of your statement "using ordinary mathematics we may arrive (or calculate)" This has nothing to do with measurement, or dense space or angels on pinheads, or anything else other than the validity or invalidity of that statement. Period. All your side-steps won't resolve that basic mediation issue. The statement is wrong and has to go. Steaphen (talk) 01:54, 18 December 2009 (UTC)

• If "We may calculate implicitly covers all orders of magnitude below the Planck length" were true, then there could be no calculations at all involving measurements. The precision depends on the precision of the starting measurements - any freshman college physical science student knows that.--JimWae (talk) 02:32, 18 December 2009 (UTC)
• I have no way of knowing what "really happens" below Planck levels. Neither do you, (though your writings seem to suggest you think you do, when you keep repeating "it is wrong") and likely nobody will ever know. We apply models until the model no longer works. Using a continous model, we do not have to discard all the laws of physics such as conservation of momentum--JimWae (talk) 02:32, 18 December 2009 (UTC)

re your "If "We may calculate implicitly covers all orders of magnitude below the Planck length" were true, then there could be no calculations at all involving measurements" -- If I am understanding you correctly, now, then your "we may calculate" implicitly means "we may approximately calculate". You either calculate precisely or approximately. Again, what exactly is it that you are calculating? Angels on pinheads? If you are accurately calculating physical stuff, then again, what happens below the Planck length?

re your "I have no way of knowing what "really happens" below Planck levels", and yet you claim you can calculate (at and below Planck levels)? You can't know what is really happening, but you believe you can nonetheless apply some mathematical/algebraic/geometric model to those realms? This has gone beyond entertaining, to surreal.Steaphen (talk) 03:48, 18 December 2009 (UTC)

So let me see if I understand you correctly. You can calculate when Achilles overtakes the tortoise? But, we can't calculate any movement if it involves sub-Planck level movements? So, that requires that when Achilles overtakes the tortoise, he "jumps" past any sub-Planck movements?

Do I understand you correctly now? Steaphen (talk) 04:54, 18 December 2009 (UTC)

So if Achilles overtakes the tortoise, say at 30 + 0.01^{-1,000,000,000} metres (i.e. at 30 metres + a sub-Planck fraction of a metre), we can't, according to your statements, accurately calculate this? But you say on the main page, "using ordinary mathematics we can arrive (calculate) ..." Which is it? We can calculate (in and through sub-Planck movements) or we can't? Steaphen (talk) 05:02, 18 December 2009 (UTC)

• As explained to you elsewhere on this page, by me & others, calculations are not what produces the lack of precision, it is the measurments we begin with. 100 metres does not mean 100.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000… metres. (This is taught in every freshman college physical science class) Have you given up on trying to read my mind now, or are you purposely misunderstanding me?--JimWae (talk) 05:20, 18 December 2009 (UTC)

You can reference "freshman", junior or elementary school mathematics/algebra or whatever. It is irrelevant, because none of them covered the issue of Planck length and below movements. NONE of them. I could say "the Earth is flat" because within certain approximations it is. We're dealing with the specifics here of how physical things move.

According to your statement above, it does indeed mean 100.0000001 or whatever, because you can calculate it precisely, to infinite degree. Either that or you cannot calculate precisely.

Again, simple question to what extent can you calculate the location and momentum of physical stuff. Forget about measurement, I'm asking you, THEORETICALLY, what is the limit of that calculation. You've said it is (implicitly) infinitely precise. I have not once seen you say otherwise. Having implicitly stated your case (since you've not denied otherwise) please find one physicist who will support it. Just one! Steaphen (talk) 06:30, 18 December 2009 (UTC)

In any event, irrespective of whatever measurements are made, theoretically, the de Broglie wavelength of an object is :$\lambda ={h}/{p}$ (where p = momentum). The infinitesimal precision of the object's position (as required by infinite-series/algebraic/geometric/mathematical solutions) requires that $\lambda$ approaches zero (since the de Broglie wavelength of the object indicates the range of possible positions and momentums of the object.). This requires the momentum to be infinite. This is nothing to do with measurement. This is quantum theory explaining the limits of theoretical knowledge.

The statement "using ordinary mathematics we can calculate' is wrong. The de Broglie relationship is most certainly important when considering movement of runners, arrows etc in the realm of Planck length increments, because to calculate them to such precision, requires they have large -> to infinite mass/momentum (depending on how 'precise' you want to 'calculate').

The mediation will not be settled unless you can find a Reliable Source who will state that exact calculation of position and momentum of physical objects is possible at and below the Planck length.

Any theoretical calculation within whatever degree of certainty (as given by the de Broglie relationship) limits the theoretical knowledge of its position. There is no way around this, unless you want to disprove the wave-nature of physical objects. The statement on the main page "using ordinary mathematics we can calculate" is deeply and comprehensively wrong.Steaphen (talk) 08:02, 18 December 2009 (UTC)

---

You keep mentioning de Broglie's wavelength. Just a simple question, does an object that has not infinite momentum a position? Ansgarf (talk) 09:00, 18 December 2009 (UTC)

Finally, you've started to ask some intelligent QUESTIONS. Do the maths (that's an abbreviation for mathematics), or algebra whatever, and starting asking some real questions as to what is really going on. So if velocity (non-relativistic) for an average size/weight runner of 90kg is say 5 m/s, that means ...

Email me when a mediator shows up. Otherwise I'll pop back in a few weeks or months to initiate arbitration.

Ciao Steaphen (talk) 20:04, 18 December 2009 (UTC)

Glad that you understood this question. I thought that if I tone it down and ask you in small steps you might be able to keep pace and provide at least some answers. Unfortunately, it seems you try to duck the question by being cute and saucy. Rather than seeing school yard antics, I would have preferred an answer. It shouldn't be hard to tell whether you assume that an object has a well defined position or not, regardless of momentum. Ansgarf (talk) 21:54, 18 December 2009 (UTC)

---

Steaphen, in case you missed my question. At or below the Planck length and time, do you think that it makes sense to ask at which point a particle is exactly, or do you believe that they are quantum superpositions? Just to put things in context, the Planck length is in the order of $10^{-35}m$, a hydrogen atom ($10^{-10}m$). This proportion is 100.000 times larger than our relation (order of $10^{0}m$) to the Milkyway ($10^{20}m$). Ansgarf (talk) 23:59, 17 December 2009 (UTC)

Dear Ansgar, I'm sorry, but I have simply lost patience or interest in your replies. They lack even a modicum of reasonable analysis. Seriously. I'm not able to offer you any replies that you seem to be able to comprehend. This latest by you is a good example. The solutions to Zeno's Paradoxes, by whatever mathematical means, whether by infinite-series, or whatever, must and do involve going through not only the Planck length and shorter increments, but infinite orders of magnitude shorter. We're not talking 10^-35 metres, we're talking 10^{-1,000,000,000,000,000,000} metres and on to infinity. Your inability to follow through with your statements and theories as to what that implies in terms of actual physical reality (e.g. the situation with runners, hares, arrows etc.) reveals a disconnect of theory with reality that I'm unable to bridge, or understand. As before, blessings on your journey. I genuinely mean that, because I can't help you, it seems, any other way.Steaphen (talk) 00:58, 18 December 2009 (UTC)

And I am well aware that you think about all order of magnitude, and I am well aware there exists an $n$ such that $1/2^{-n}$ will be smaller than any positive $e$. I do know the definition of convergence. And it is in all orders of magnitude a grave category mistake to apply uncertainty to calculations, because calculation work on dimensionless numbers. In the set of real numbers with addition it is true that 1+1 is exactly 2. And 100 +100 = 200. And $1*10^{-40}+1*10^{-40}=2*10^{-40}$. This follows from the simple fact that addition and multiplication on the reals is distributive. And no physicist assumes otherwise.

You said repeatedly that you "believe physical stuff (...) exists (at and below the Planck level) in superpositions of possibilities". But you still ask Jim and others what the exact position of physical stuff is. If there is a disconnect, then between what you claim to believe, and what your question and remarks reveal you actually believe. The statement about the relative size of hydrogen and the planck length, compared to you and the Milky Way, was intended to make you think about the concept of position. Because you keep asking at what postal address the Milky Way resides. Figuratively, speaking.

You lost interest, because you have no reply, and probably also because you cannot stand exacting analysis. If my analysis looks absurd, then because it starts on purpose from some of your assumptions and statements, and reduces them to absurdity. Which is made easy, since I have seen very little evidence that you understand what it means for a set to be dense, what is meant by superposition, what model of time is used actually in QM, the nature of mathematical proof, the nature of physical models, the role of experimental evidence, or what assumption Zeno makes, just to name a few things. In your latest reply you just try to duck that you haven't thought about the fact that in the order of Planck levels, the naive notion of position of a particle may not apply. Or you only think about it if it suits you. Ansgarf (talk) 18 December 2009 (UTC)

Significant figures

Strictly speaking, 100 metres has only 1 certain significant figure and indicates any distance from 50 metres to less than 150 metres. To clearly signify the 3 significant figures usually intended by 100 metres, we could write instead the odd-looking "100. metres", indicating from 99.5 to less than 100.5 metres. "100.0 metres" indicates from 99.95 metres to less than 100.05 metres - that the value is less than one that would round to 100.1 and greater than one that would be rounded to 99.9. Standard 100-metre races are probably exact, at best, to 0.0005 metres (1/2 a millimetre). Even if exact to 1/20 of a millimetre, we should only specify the distance as 100.0000 metres. People who work in the physical sciences are expected to be aware of the limitations of all measurements, and avoid reporting with false precision. QM cannot be the primary cause of uncertainty when the measurements are at this level of precision. The main uncertainty comes from the fact that speed, distance, and time are measurements, not ideally exact numbers. Though there are less clear standards for fractions, 11 1/9 seconds indicates a time (in seconds) greater than what would round to 11 2/17 and less than one that would round to 11 2/19. If I thought the fractions were the reason for Steaphan's concerns, I would have used values that could be presented in decimal form long ago. Anyway, it does not matter how precise the figures in the example are, if it mathematically shows that Achilles will actually pass the tortoise. --JimWae (talk) 09:37, 18 December 2009 (UTC)

Not quite, "100" is simply ambiguous. With physical quantities, you have the option of using SI prefixes, as in 100±0.5 m = 1.00 hm or 10.0 dam. Regards, Paradoctor (talk) 19:29, 27 December 2009 (UTC)

But "100", unless otherwise specified, cannot be presumed to have more than 1 sig fig.--JimWae (talk) 22:56, 28 December 2009 (UTC)

That's what "ambiguous" means, you cannot presume. It may mean 1 hm or 1.00 hm, or if you prefer km, either 0.1 km or 0.100 km. Paradoctor (talk) 00:29, 29 December 2009 (UTC)

Stillness

It's rather strange. Here we are considering how motion is possible, when the present-day view is that stillness is impossible - at both the macro & sub-micro levels--JimWae (talk) 01:48, 18 December 2009 (UTC)

Do you realize that rest is an extreme case of motion? ;) Paradoctor (talk) 21:27, 12 February 2010 (UTC)

This page is crap

I read this entire page of arguments and its is crap, you guys are nerds who need to drink some beer, no one should care about philosophy this much when you will die one day. —Preceding unsigned comment added by 173.26.222.43 (talk) 11:20, 21 December 2009 (UTC)

My what a talk page. Is this guy Steaphen attempting to illustrate the paradox with infinitely recursive argument? --77.188.52.212 (talk) 16:45, 27 December 2009 (UTC)

Nope. I'm waiting for him to start arbitration. Paradoctor (talk) 17:38, 27 December 2009 (UTC)

No point waiting. You are not involved in this mediation (come arbitration)-- your comments and opinions are not relevant or required.Steaphen (talk) 20:01, 22 January 2010 (UTC)

(sipping tea) Paradoctor (talk) 22:04, 22 January 2010 (UTC)

Proposed changes

I propose the following changes:

1. Paragraph "Zeno's paradoxes were a major problem .... wrong with the argument."

I propose to replace this with

"Zeno's paradoxes were a major problem for ancient and medieval philosophers. More modern calculus has solved the mathematical aspects of the paradox, while many philosophers still hesitate to say that all aspects paradoxes are completely solved. Variations on the paradoxes (see Thomson's lamp) continue to produce philosophically and mathematically challenging problems. Developments in physics have called into question the idea that position, time, and speed are simple points, which undermines some of the implicit assumptions of Zeno paradox."

The reason for this change is that the previous version puts mathematics and physics needlessly in opposition. Physics is unrelated, and plays it own role. Also, the previous version does not explain its role.

2. Paragraph "Using ordinary mathematics (...) namely, "How is it that motion is possible at all?""

I propose to delete the entire paragraph. Zeno's paradox is not about algebra, or how to compute when two objects meet, or how to compute where two lines in Euclidean space intersect. This paragraph is also too specific for an encyclopaedic article.

3. I propose to not qualify the word "calculate". Within the scope of the paradox positions, times and limits of series are not "approximately calculated", or "more exactly calculated", but simply "calculated". The article is not about uncertainty, robustness, error bounds or numerical accuracy.

4. The paragraph "Physicists remark (...) about 10−16 seconds."

I propose to delete the last sentence "As of 2004, the shortest time difference capable of actually being measured was about 10⁻¹⁶ seconds." It is not relevant for the paradox, and this information should be mentioned in the article on the Planck length.

I propose to replace it with the sentence. "These findings suggest that for physical systems the infinite series that appear in Zeno's paradoxes may not occur at the sub-quantum level." Ansgarf (talk) 04:54, 29 December 2009 (UTC)

Mediation

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

This is not the appropriate section for mediation. The mediation page is at http://en.wikipedia.org/wiki/Wikipedia:Requests_for_mediation/Zeno%27s_paradoxes

Steaphen (talk) 13:19, 15 January 2010 (UTC)

I know :) but since im on medcom, I'm asking whether there is still a need for a mediator :) Seddon ^talk|^WikimediaUK 04:30, 31 January 2010 (UTC)

The mediation issues in the main have not been resolved. If you would like to make relevant representations on the appropriate pages, that would be appreciated. I expect you've noted that the POV notice in the main article was, in violation of Wikipedian guidelines, removed without the issues being resolved. Hence the likely need for arbitration, given the persistence of bias and POV in the main article.Steaphen (talk) 08:48, 1 February 2010 (UTC)

Arbitration called

Wikipedia:Arbitration/Requests/Case#Zeno.27s_paradoxes Steaphen 13:37, 11 February 2010

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

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

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

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

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

Outside opinion by CBM

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

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

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

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

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

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

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

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

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

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

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

I agree with the thrust of CBM's comment, but with regards to "naiveté" consider the following:

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance... Bertrand Russell, The Principles of Mathematics (1903)

Paul August ☎ 18:58, 11 February 2010 (UTC)

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

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

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

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

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

I agree, completely. Boris Tsirelson (talk) 22:16, 11 February 2010 (UTC)

Just as a reminder. The current article does mention the philosophical aspects as well as the mathematical aspects of the original paradox, and that more than once. It also mentions that when applied to physical reality quantum aspects may play a role. These paragraphs may be improved, but it is not the case that the current article doesn't mention it. It does.Ansgarf (talk) 23:27, 11 February 2010 (UTC)

Yes, you're right. I read through the comments higher on the page, and I thought that both sides were right, but about different things. The lede section of the article has a nice tone, I think. The difficulty is the section "Status of the paradoxes today" lower down, which tries to do too many things at once. I would propose adding a section "mathematical resolution" that discusses the calculus-textbook approach, and then editing "status of the paradoxes" to focus more on the philosophical reception of the mathematical solution. Looking through google books, I am sure there are enough sources to do all of this. — Carl (CBM · talk) 00:51, 12 February 2010 (UTC)

A textbook solution just got removed a few weeks ago, since too many people felt that the paradox is at its core not about calculus or algebra. In my opinion it wasn't worth the trouble to be included [3]. But if you can come up with a paragraph that that avoids the problems the old version had, I wouldn't object.

Mentioning Brouwer at that point was a compromise, on an earlier formulation. It claimed that Intuitionists reject to any use of infinites, which is not true. [4]. The most recent sentence on Brouwer was intended to keep those people happy, while at the same time keeping them from adding false claims about Intuitionists. I agree that the whole topic is a bit spurious, but it is a compromise. Ansgarf (talk) 01:14, 12 February 2010 (UTC)

I see now. Regarding the calculus solution: to many mathematicians, including me, the calculus solution "is" the solution of the paradoxes. It's certainly repeated as a solution in numerous calculus and analysis texts. Moreover, I saw several sources today that seemed interested in discussing whether it really was a solution. I think the article would be incomplete without covering the mathematical viewpoint. On the other hand, I understand the philosophical viewpoint that the mathematical solution is too idealized to address the original problem.

Regarding Goedel and Brouwer, as a logician I think discussing them at all is a red herring. The issues that the intuitionists have with infinity (and you are right that they do not reject it outright) are not related to infinite sums nor to classical mechanics, while Zeno's paradoxes are unrelated to the law of the excluded middle. Goedel's theorem is not related to the foundations of calculus, which were accomplished in the 19th century anyway, well before his time. — Carl (CBM · talk) 01:42, 12 February 2010 (UTC)

Ok, maybe there is something to be said to leave out everything that is too tangential. And Intuitionism probably is. If you can include a short treatment of the mathematics of the paradox - which might already be achieved by reorganising the current article a bit - then I'd be more than happy. The article does are ready contain references that calculus is not "the" solution to all aspects. By some reorganisation this might become more pronounced. Ansgarf (talk) 02:30, 12 February 2010 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Yes, it's really not that hard to understand: every theory "cannot be substantiated via physical experimentation or evidence", if you require it to be absolutely right. Does it mean that we should stop using theories at all, however? Boris Tsirelson (talk) 08:09, 12 February 2010 (UTC)

When the article talks about "While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise, ..." it refer to the tortoise and the runner in the mathematical model of the paradox. To solve the equations you are not doing "experiments", you do math. Which will give you mathematically precise solutions. That is much more rigorous than any experimentation can be. They are not any particular runners or tortoises that have a shoe size, a weight, or feeding habits. That Zeno's paradox is a thought experiment, a proof by contradiction, and not an actual physics experiment, is obvious from the context, and even mentioned explicitly in the article. Ansgarf (talk) 08:25, 12 February 2010 (UTC)

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

It is not a problem to overtake the Tortoise (as Zeno surely understood). It is a problem, whether a finite time interval can contain infinitely many events, or not. Boris Tsirelson (talk) 10:05, 12 February 2010 (UTC)

As Carl emphasizes, we should take into account both sides of the story: (a) some models resolve the paradox admitting infinite divisibility of space-time (the mathematical side); (b) it does not mean that infinite divisibility is a property of reality (the philosophical side). Let me repeat Carl's phrase: "Of course these two approaches (mathematical/philosophical) are not in conflict. They complement each other by revealing different aspects of the situation." Boris Tsirelson (talk) 10:10, 12 February 2010 (UTC)

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

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

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

(Yes, I realize that this does not even come close to doing justice to the philosophical side of this near-2500-year-old-and-still-kickin' paradox; however, it does clearly show that those seedy, unphilosophical (aphilosophical?) mathematicians are certain that Zeno's paradox has been "solved".)

 —  Paine (Ellsworth's Climax)  11:36, 12 February 2010 (UTC)

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

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

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

But what about the solution (to the dispute, not to the paradox) proposed by Carl? Does it satisfy you? Any objections? Recall it: "Of course these two approaches (mathematical/philosophical) are not in conflict. They complement each other by revealing different aspects of the situation." OK? Or not? Boris Tsirelson (talk) 12:19, 12 February 2010 (UTC)

If I am not mistaken Carls proposal is to state that "(a) some models resolve the paradox admitting infinite divisibility of space-time (the mathematical side); (b) it does not mean that infinite divisibility is a property of reality". I am happy with this distinction, and I assume most people are, since this distinction is already reflected in the article, and has been in there for a long time. Ansgarf (talk) 14:07, 12 February 2010 (UTC)

Okay, Steaphen, I shall continue to look for reliable sources. I realize that WP sometimes frowns upon YouTube as a source, however since the content guideline clearly states that there is no blanket ban against YouTube, and since the LINK I gave above is a video made by a reputable professor of mathematics at U. of Helsinki and Florida State University, perhaps then we could begin with THIS LINK as a reliable source and an inline citation? That web page prominently links to the YouTube video I cited above. Please keep in mind that it is not our job to debate the TRUE vs. FALSE, the RIGHT vs. WRONG, SCIENCE vs. PSEUDOSCIENCE, etc. of any reliably sourced claim. All we must do is agree that the source(s) is reliable. So do you accept

THIS LINK 

as a reliable source for the claim:

"While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise . . ."

 —  Paine (Ellsworth's Climax)  13:36, 12 February 2010 (UTC)

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

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

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

It is not up to me, and it is not up to you whether or not the claim is valid. The only thing that we editors get to decide is whether or not a claim can be reliably sourced. And the claim about the mathematics that, in the eyes of mathematicians resolves Zeno's paradoxes is a valid claim and can be reliably sourced.

 —  Paine (Ellsworth's Climax)  15:55, 12 February 2010 (UTC)

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

Great! Perhaps you can slip one or two of those PPTs after the claim in question? And just to provide a focus in case it might be needed, I'll place a {{cn}} template in the section of the article where the reliable source is called for by Steaphen.

 —  Paine (Ellsworth's Climax)  19:16, 12 February 2010 (UTC)

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

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

Naïve question

Is this helpful? To wit, does anybody doubt that Achilles will overtake the tortoise at

$t_{\text{pass}}={\frac {l_{\text{tortoise}}}{v_{\text{Achilles}}-v_{\text{tortoise}}}}$

and

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

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

I don't doubt it, Paradoctor, however when another editor sees the need for a reliable source to support a claim, then I don't see how this can be ignored. I placed the cite-needed template at the precise place in the text that follows the claim that's in dispute just in case any other editors wanted a quick focus. Please note that I've checked the three cites that follow the next word, "Philosophers", and they do not appear to support the math claim.

I do not have access to, nor would I understand very well, the calculus text(s) that would make good, reliable sources, or I'd do it myself.

 —  Paine (Ellsworth's Climax)  20:33, 12 February 2010 (UTC)

'three cites that follow the next word, "Philosophers"': They are not meant to, they relate to the following claim.

As expected, JimWae came up with sources supporting my contention. The challenge has been met. Or does anyone have any source proposing alternative values for $t_{\text{pass}}$ and $l_{\text{pass}}$, Steaphen? Apart from suggesting that physics is so fundamentally wrong that it overlooks the impossibility of motion, of course. Paradoctor (talk) 21:22, 12 February 2010 (UTC)

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

 —  Paine (Ellsworth's Climax)  22:50, 12 February 2010 (UTC)

Here are some text sources to choose from: --JimWae (talk) 20:38, 12 February 2010 (UTC)

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

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

Yes! definitely a "No". Jim, if I had to choose from these best of the best (hard choice, truly), I'd opt for the first one. No, No, don't sweat it, I'll add it in, and you and others can improve it if you feel the need. Thank you very much for your effort and time!

 —  Paine (Ellsworth's Climax)  21:54, 12 February 2010 (UTC)

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

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

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

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

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

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

Okay on the vid, Jim, I understand. As for conflicts and paradoxes and math, all I can say is that if the arrow really can't move, then we are all being tricked by one hell of an illusion!

 —  Paine (Ellsworth's Climax)  21:55, 13 February 2010 (UTC)

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

Grünbaum

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

Bibliography

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

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

• Grünbaum, Adolf (1967). Modern science and Zeno's paradoxes. Wesleyan University Press. Retrieved 13 February 2010.

• Grünbaum, Adolf (1968). Modern science and Zeno's paradox. Retrieved 13 February 2010.

• Salmon, Wesley C. (March 2001). Zeno's paradoxes. Hackett Publishing. ISBN 9780872205604. Retrieved 13 February 2010.

Salmon's book is one of the best on the subject. Huggett, in his article "Zeno's Paradoxes" in the Stanford Encyclopedia of Philosophy [5] writes: After the relevant entries in this encyclopedia, the place to begin any further investigation is Salmon (2001), which contains some of the most important articles on Zeno up to 1970, and an impressively comprehensive bibliography of works in English in the Twentieth Century . Paul August ☎ 14:22, 13 February 2010 (UTC)

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

• Alper, Joseph S.; Bridger, Mark (1997). "Mathematics, Models and Zeno's Paradoxes". Synthese. 110 (1): 143–166. doi:10.1023/A:1004967023017. ISSN 0039-7857.

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

• Sewell, Kip K. (1 October 1999). The Cosmic Sphere. Nova Publishers. ISBN 9781560726616. Retrieved 1 March 2010.

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

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

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

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

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

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. — 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. — 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

1. attributable (per WP:A)
2. 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.

• 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.

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)

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. — 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 [6]. 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 [7] and [8]. 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 (UTC)

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.

 —  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

I've made a bold edit 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?

 —  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.

 —  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.

 —  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.

 —  Paine (Ellsworth's Climax)  12:29, 23 February 2010 (UTC)

Allow me to reiterate 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.

 —  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!

 —  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 Wikipedia: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.

 —  Paine (Ellsworth's Climax)  22:45, 25 February 2010 (UTC)

How about including Peter Lynds paper as source [9]. I would however keep in addition the textbook source, and add another textbook [10]. 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!

 —  Paine (Ellsworth's 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.

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.

The screen below is blank, use it. (The order of the sections is alphabetical, nothing else) Paradoctor (talk) 13:48, 22 February 2010 (UTC)

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 {{cn}}ed 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" [11]. 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 {{cn}}ed. 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?

 —  Paine (Ellsworth's 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)

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.

 —  Paine (Ellsworth's 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)

 Done, and thank you very much, Steaphen!

 —  Paine (Ellsworth's 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.

 —  Paine (Ellsworth's 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 #On_the_subject_of_reliable_Reliable_Sources - 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))

 Done

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?

 —  Paine (Ellsworth's 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:

1. "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?
   
   • Proposed change: "More modern calculus has solved, with high accuracy, the mathematical aspects of the paradox."* Done.
3. "while many philosophers still hesitate to say that other aspects of the paradoxes are completely solved"
   1. Which philosophers 'hesitate'? And what about those that don't hesitate in the least, but state categorically calculus does not solve the paradoxes?
   2. 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 (Ellsworth), which ties 'fair accuracy' to any such 'partial solutions'.
• Proposed change: remove the prejudicial 'hesitate' and replace with "Many philosophers argue that, beyond any high-accuracy solutions, the paradoxes remain unsolved."
4. What reliable Reliable Sources confirm that the paradoxes are solved beyond 'fair accuracy', and what is the specific experimental evidence supporting that claim?
5. "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?
6. "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?
7. "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'
8. ...more, soon. Steaphen (talk) 03:20, 27 February 2010 (UTC)

• 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.

• "Modern calculus has solved the mathematical aspects of the paradox"

No need for change. You can solve mathematical aspects with mathematics.

• "while many philosophers still hesitate to say that other aspects of the paradoxes are completely solved"

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:

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 [12] it is for example postulated that $e=mc^{2}$. 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" [13].

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 [14]. See the 2009 proceedings as example [15].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

• Revert to the old version
• Omit the "highly accurate"
• Or change it to Many philosophers argue that beyond mathematical solutions the paradoxes remain unsolved.

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)

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." -- 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[16].

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)

Rucker

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• Further reading: [17]
• Mentioned alongside Bertrand Russell: [18]
• Referenced: [19]– [20] (in Peter Suber's bibliography)– [21] (scholar search of Google)

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

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

Changes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thank you. I have, to my knowledge, never disallowed or dismissed the technical mathematical validity of anything on this talk page, only its direct relevance to the issue of physical movement (aka the problems posed by Zeno of Elea). By all means, fiddle with your slide rules and come to whatever conclusions you like, but PROVE that calculus actually correlates with what you say it does in relation to physical movement through ALL scales of movement. If you can't prove it, the 'high-accuracy' qualifier remains the most scientifically credible and testable! Steaphen (talk) 03:38, 2 March 2010 (UTC)

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

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

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

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

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

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

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

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

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

In my comment I meant to say that I can live with the "wrong version" until we got feedback from others. No we have feedback from others. Ansgarf (talk) 03:41, 2 March 2010 (UTC)