Talk:Zeno's paradoxes/Archive 3

Explanation for my recent edits
My recent edits were reverted by. I did not think those edits would be contentious, which is why I didn't seek a discussion here first. No disrespect was meant, and I hope nobody has taken offense.

Now, regarding my edits:
 * 1) I removed the statement "while developments in physics have called into question the idea that position, time, and speed can ever be exactly determined" from the lead. Per WP:LS the lead section should be a summary of the article. The article nowhere mentions these "developments of physics" (quantum indeterminacy, I assmue?), nor&mdash;more importantly&mdash;what these developments have to do with Zeno's paradoxes. If it should remain in the lead the article should at least contain some mention of how quantum indeterminacy relates to Zeno's paradox.
 * 2) I removed a paragraph on Peter Lynds. Lynds represents the fringe of science, and mentioning him in this article seems like a gross violation of the policy regarding undue weight.
 * 3) I removed a sentence tagged for over a month.
 * 4) I removed a quote by Bertrand Russell. The quote doesn't seemingly relate to Zeno's paradox, merely to ancient paradoxes in general.
 * 5) I removed a BBC article relating to the shortest time measured. Again, per WP:EL, the external links should be "directly relevant", and the BBC article mentioned is at best indirectly relevant to this article.

In hindsight, I yield that actions 3 and 4 might have been premature. But I'm interested in anyone's reason for opposing 1, 2 and 5. Gabbe (talk) 09:13, 4 September 2009 (UTC)


 * In regards to point 1, as explained in the mediation, there is (and remains) a bias towards the assumption of continuity (of movement, physicality) and needs to be removed, unless a reliable source confirms the assumption is valid. In view of the fact that informal mediation failed to elicit the necessary reliable sources (see above), formal mediation will soon be initiated to correct the errors in the article.Steaphen (talk) 23:31, 5 September 2009 (UTC)

There have been for some time 3 or 4 reliable source cited in the article that discuss the position that space &/or time need not be construed as continuous. They are presently in the Status Today section but properly belong as as Proposed solutions also. As such, "smallest measurable" time is relevant, as would be a discussion of the different ways (some jump to the conclusion it is also the smallest "meaningful" unit) it could be understood. I agree with removal of Lynds - not because it is fringe but because it is not unique to him. The idea that there is uncertainty in assigning positions & times is important to understanding how to deal with the ZPs, whether it be a knock-down argument or not--JimWae (talk) 02:33, 6 September 2009 (UTC)


 * First of all, I'm not saying the BBC article is blatantly irrelevant, merely that it is not directly relevant. The external links section is for providing links to external pages which are directly relevant to the article topic, not those that simply are "important to understanding how to deal with" the article. See WP:ELNO #13: "Sites that are only indirectly related to the article's subject: the link should be directly related to the subject of the article." As we seem to agree on Lynds at least, I'll delete that. Gabbe (talk) 06:28, 6 September 2009 (UTC)

The sections of the article that need improvement are the "Proposed solutions" and "Status of the paradoxes today". I think wikipedia cannot determine what the "current status" is & these 2 sections need to be combined. Removing an external link that neds to be included somewhere in the article does not address what the article sorely needs. Solutions have been proposed by "the ancients", "math & calculus", and by modern physics. Modern physics calls into question the assumtion that space & time are infinitely divisible. Relevant to modern physics would be the uncertainty in all measurements, Planck units, & the smallest distances & times so far differentiated. Kantian considerations call into question whether space & time are things that can be divided at all. I will be working on something along these lines. I will not conclude that any proposed solution is entirely right or entirely wrong. --JimWae (talk) 18:39, 7 September 2009 (UTC)


 * I dispute your claim "being there 4 years indicates general acceptance", made in this edit summary. Look at WP:BAD for example. There are several total hoaxes that have survived for several years, so the fact that something has remained in an article for several years does not mean that there's a wide acceptance for keeping it.
 * I wonder, could you explain why you think the BBC article is directly related to the article topic? Gabbe (talk) 21:15, 7 September 2009 (UTC)
 * I've asked for further opinions on Wikipedia talk:WikiProject External links and WP:CNB. Gabbe (talk) 21:59, 7 September 2009 (UTC)

I do not think off-topic properly described the link, but it did need to be better connected to the text of the article. I have done that & made the link a ref for some article text. I resited removing the link, because I did not think that just removing material was going to do what needs to be done to improve the article. There is yet more to be done --JimWae (talk) 07:15, 8 September 2009 (UTC)


 * Ah, great, the "external links" thing is behind us and we can move on. Gabbe (talk) 07:59, 8 September 2009 (UTC)

The paradox with physics
There seems there was quite remarkable discussion going, and I noticed that it had unfortunately influence on the content of the article. While Zeno's paradoxes are thought experiments, there are apparently participants in the discussion who claim that physical reality determines what Zeno can imagine, i.e. since it might be impossible to measure time and space below a certain threshold, Zeno could impossibly imagine a point in time or space between two points below this threshold. This is an extreme form of materialism - you can only imagine what exists - that I haven't come across often. So, I am pleased to have learned something new.

To deal with the current discussion the paradox we don't need to get all philosophical. However, before I say a few words about the paradox, first a remark on terminology. To define the first half of Zeno's paradox - about the sum of distances - you do not need that the line between Achilles and the tortoise is "continuous". A dense line is sufficient. Zeno only assumes is that there exists a point half way between two points  that Achilles has to cross. The rational numbers are more than sufficient for this purpose and there is no need for the real numbers. But his is just a remark aside.

Now, lets look at the paradox itself. There are two parts to the paradox of Achilles.


 * In the first part Zeno argues that you can construct an infinite sequence of distances, and that if you add them up the sum is still bounded. He argues that the infinite sum is less or equal to the distance to the tortoise.


 * For the second part Zeno argues that to cross any of the distances, Achilles needs a positive amount of time. He then claims that an infinite sum of times is unbounded. Hence the paradox.

Calculus provides a solution to this apparent paradox by pointing out that the infinite sum of times is actually bounded as well. And it turns out that the infinite sum of times is actually the time that Achilles needs to cross the overall distance.

Stephean is right, calculus assumes that time is dense, i.e. that in between every two points in time you can find another point in time. And he is also right to point out that in reality space/time might not be dense, but that there might exist a smallest physical distance/time. Furthermore, if there exists a smallest distance/time, then an infinite sum of non-zero times will be unbounded. Stephaen is right about this, and this is just what Zeno assumed implicitly as well.

However, if there is a smallest distance/time - and this is what Stephean and others miss - then it holds equally true that the infinite sum of distances is unbounded. And certainly larger than the distance to the tortoise. Hence, Achilles can pass the tortoise easily.

To summarise. If space/time is dense then calculus tell us that both the infinite sum of distance and time are bounded. And the paradox avoided. If space/time is not dense, and there exists a smallest distance between between points in space/time, then neither of these sums is bounded. And hence the paradox avoided as well.

I propose to change the entry accordingly, especially the part that talks about physical time and distance and its significance for the paradox. Ansgarf (talk) 09:33, 23 October 2009 (UTC)


 * Zeno never "argues that you can construct an infinite sequence of distances, and that if you add them up the sum is still bounded". A Zenoite/Parmenidean wants to show that assuming motion leads to a reductio ad absurdum, and would contend that "in reality" 'all is one' & distance, time, and motion are illusions - and that Achilles & the tortoise are not any distance apart to begin with. He also is arguing that the (illusory) task of catching the tortoise requires an infinite number of subtasks - and no math can make an infinite number of discrete actions take a finite time. Zeno maintains there are an infinite number of actions/movements involved in catching the tortoise. Calculus can find the sum of a infinite sequence of diminishing numbers, but it cannot find a way to complete an infinite number of tasks even if those numbers are meant to model what is happening. --JimWae (talk) 02:31, 24 October 2009 (UTC)
 * Can you make up your mind. Does Zeno divide the task of overtaking the tortoise in an infinite number of movements of non-zero lengths, or does he not?


 * If he does not, then there is no paradox. And if he does, does he argue that these sum movements falls short of adding up to the distance to the tortoise, or does he not? If he wouldn't, there wouldn't be a paradox.


 * If most of these smaller distances do not exist in reality but are illusory, then also the associated task is only illusory as well. And if he considers all of the tasks to be actually tasks you have to complete, you have to ask for the reason why you cannot complete an infinite number of tasks? The associated durations of the tasks do add up to a finite time, so there is no reason why you shouldn't be able to complete them in a finite time. Or why shouldn't you be able to complete an infinite number of discrete actions in finite time? Ansgarf (talk) 03:04, 24 October 2009 (UTC)


 * 1>An infinite number of tasks can never be completed - there are always more tasks to do. Likewise, we cannot finish adding an infinite number of numbers - even though we can sometimes find the sum by another method. 2> He is saying that assuming there is motion, distance & time leads to a contradiction that makes motion impossible. I don't buy his conclusions, but that does not matter.--JimWae (talk) 06:34, 24 October 2009 (UTC)


 * I get the form of his argument. Zeno tries to prove his point by reducing the opposite claim to a paradox. The question is why should it be impossible to complete an infinite number of tasks. I admit that you cannot "add" an infinite number of numbers in the naive way by adding them one by one, but its only impossible, if we assume that "adding" takes a minimal amount of time. Achilles however doesn't "add" i.e computes anything, he "moves". And moving over smaller and smaller distances takes smaller and smaller times. So, why should it be impossible to complete an infinite number of (imaginary) tasks? Give me the reason. Ansgarf (talk) 07:00, 24 October 2009 (UTC)


 * It's quite simple: an infinite sequence of numbers has no last number & an infinite number of tasks has no last task --JimWae (talk) 07:44, 24 October 2009 (UTC)
 * That is quite true, and implied by the word infinite. But why shouldn't you be able to complete them? You seem to handle the definition that a list of task is completed at time t, if the last task is completed at or before time t. And that definition only applies to finite lists, as you correctly noted. But an alternative definition and evenly valid definition is: A list of tasks is completed at time t, if each task is completed at or before time t.
 * Now, the tasks or movements of Achilles can be numbered 0,1,2,3,4, ... . Each task has a unique number n. Let's assume that task number n starts as time 1-1/2n and stops at 1-1/2(n+1). This way it is actually quite easy to squeeze an infinite number of tasks into a finite amount of time. Or can name me a single task in this list that is not completed at time t=1? Ansgarf (talk) 12:54, 24 October 2009 (UTC)


 * Again, that is using a different method - one that assumes that time t=1 will eventually "happen". That is not an assumption Zeno would allow, no less argue for. It does support the argument given by Aristotle, and would be a good addition to the proposed solutions in the article, but it does not defeat Zeno's argument. It does not even require calculus (nor do several other arguments already included).
 * If we add {0.5, 0.25, 0.125, ...) through the first 15 elements we get  0.999969482421875000...(repeating). The more we continue, the more apparent it is that eventually every decimal place will become a 9, and that the sum of the entire set will become 0.9999999999...(repeating). It can also be proven that 0.99999999...(repeating) exactly equals 1.0... Again, a different method - one that relies on our understanding of what can & cannot happen.
 * I think we cannot assume that we can say anything about the "nature" of space & time. These are unavoidable constructs we have that shape our understanding, but the terms do not need to refer to any entities, much less entities with "properties". We can model space & time as Euclidean or non-Euclidean, discrete or continuous, depending on which model is most appropriate for the task at hand. There is no "correct" model, there are useful ones, and all have their limitations.--JimWae (talk) 19:39, 24 October 2009 (UTC)
 * First, 0.99999.... (repeating) is 1, and that doesn't follow from any complicated methods, but simply how floating point numbers work. Foating point number 0.99999.... is the floating point way to write down 9/9.
 * Second, the series 0.5, 0.75, 0.875, will never become 0.999999..... So, rather than claiming that "eventually every decimal place will become a 9", it is more correct to say that "every decimal place will become eventually a 9". Small but significant difference. So, as you see, I agree that the series never reaches 1.
 * However, neither does your series of tasks include a single task that involves reaching the tortoise. It is rather unfair to define a list of tasks that does not include a task of reaching the tortoise, and then complain that if all of these tasks are completed the tortoise wasn't reached.
 * Next, of course Zeno argues that t=1 will not happen. But for what reason? And the argument is, to use your own words, that "no math can make an infinite number of discrete actions take a finite time". But of course, an infinite number of actions can take a finite time. It seems that your intuition tells you to add all these numbers one by one, and that this would take an infinity. And that is true, if each task has a minimal start-up cost. But with movement, there is no such minimum - in the dense time model. So, the intuitive notion that "no math can make an infinite number of discrete actions take a finite time" is simply not true.
 * Finally, you are right it depends on your model of time. And our discussion was using a dense time model. However, my comment on the current entry concerned the part that was assuming a discrete model in which time and space have a minimal distance. And that contrary to what the entry currently says, if time and space are discreet, then Zeno's paradox is no paradox at at all, because neither of the infinite sums exists. Or, alternatively, both of them are unbounded. Not sure if you have any objection to that? Ansgarf (talk) 23:30, 24 October 2009 (UTC)

October 2009 (UTC)
 * Could you locate for me more clearly the part you want to revise?--JimWae (talk) 06:58, 27
 * First I think the language mentioning mathematicians should be less non-committal. In mathematics the problem has been settled, full stop; and  not just to the satisfaction of some mathematicians - even though one might object for philosophical reasons to infinite series. But my main concern is with the sections that mention QM and relativity.  You worked on the Rucker quote and I agree that the quote is somewhat weird. This holds similarly for the paragraph that starts with "Phycisist point out ...". The facts it mentions have no clear relationship with the paradox, and the paragraph itself does even mention the paradox either. This is fair enough, because imho there is almost no relevance to the paradox, but it makes me wonder if we need the current paragraph at all. Given that there are people who can't resist to bring up QM, we should keep mentioning it. But mention it as having little relevance for the paradox. Ansgarf (talk) 10:32, 27 October 2009 (UTC)


 * I think that mathematicians are indeed quite committed to the claim that the sum of an infinite number of terms can be finite. But everyone else agrees with that as well. This is not the issue. The issue is whether an infinite sequence of tasks can ever be completed. As far as I can see, most mathematicians do not address this issue at all. And, if they do, I do not believe that there is any consensus. Joseph Mazur is a mathematician and in his book Zeno's Paradox he readily admits that the paradox is quite a conundrum. I am also afraid that we may well be in a situation where we have reached rock bottom in the sense that there are no arguments to support either position without making a circular argument. For example, the typical argument for doing an infinite number of tasks in a finite amount of time is to do each task at half the time as the previous one, so that if the first task took 1 second, after 2 seconds we'd have done an infinite number of tasks. The problem with this argument is that it has to assume that time is dense so as to be able to associate particular tasks with particular points in time, and so then to say that time can reach the 2 second point is to claim (without any further argument) that an infinite sequence (this time of time points) can be completed. Likewise, those that maintain that an infinite sequence cannot be completed have no further argument to back up their position other than their intuition that infinity is just that: you can't sequentially go through all the members of an infinite set and hope to ever be able to finish. To them, this is just by definition of infinity, and asking any further why's is like asking why the successor of 1 is 2. —Preceding unsigned comment added by 67.248.254.112 (talk) 13:21, 27 October 2009 (UTC)
 * Zeno's paradox assumes indeed that time and space are dense. And so does the calculus solution. The paradox amounts to the question whether you can fit an infinite number of tasks into a finite amount of time, and no mathematician disagrees with the fact you can do this in a dense time model. This is not circular reasoning, it is starting from an assumption - a dense time model - and using that assumption in the proof. For this model, there exist no argument why you couldn't complete an infinite number of tasks in a finite time. That argument exists for a discrete time model, but not for a dense time model. Of course, it is fair to ask if the dense time model is appropriate for reality, but that is a different question.
 * You can start from an alternative assumption, that space and time are not sense. If time and space are not, you couldn't divide the distance into an infinite sequence of smaller distances. No mathematician will tell you that you could. And with an infinite sequence of distances there wouldn't be a paradox to begin with. So, from the mathematics point of view there is no problem, even though some mathematicians might have philosophical objections to a dense time model.Ansgarf (talk) 23:51, 27 October 2009 (UTC)
 * Yes of course the assumption in Zeno's reasoning is that space/time is dense, we are all agreed on that. However, we disagree on whether the 'paradox amounts to the question whether you can fit an infinite number of tasks into a finite amount of time'. In saying that this is what the paradox amounts to, you are effectively saying that the paradox is simply asking whether the sum of an infinite number of terms can be finite, and if that's the question, there is indeed no paradox: we all agree that you can fit an infinite number of terms/intervals/points in a finite space/time. But I don't see that as the central question in Zeno's paradox: his question was about the whole *dynamics* of the situation: as you go through the infinite sequence one by one, how can you ever reach the end? To me, that is a question mathematics has never addressed, and so what you are doing I feel is to set up Zeno's argument as a straw man: please read the article where it makes a very similar point about how Zeno's reasoning is easily misrepresented. If you replace the part '... and since it takes an infinite amount of time to do this ...', and instead simply say '... and since you can't finish an infinite sequence ...' near the end of the argument, the whole ball game changes. With the former, it indeed becomes a dumb line of reasoning, but I have to believe that Zeno meant the latter. I mean, if he starts with a finite distance between A and B, and then divides that up into an infinite number of sub-intervals, then it is immediately clear that an infinite number of terms can add up to a finite amount: why the heck would he believe that in time this would suddenly be any different? And regardless of what Zeno himself actually believed, it is the latter argument that creates the real paradox and that needs to be addressed.
 * I wonder if maybe the following helps. I think that what we are dealing with here is that there is a contradiction (hence the paradox) between two lines of reasoning: one that says: 'A, therefore P', and another that says 'B, therefore not P'. To those that say 'P', the latter people say: 'but wait, B clearly indicates not P'. To which the former people say: 'Well, I don't see what your problem is: clearly A shows that P'. To which the latter people say: 'Look, you're not addressing the point: B shows not P'., etc, etc. In order to break this cycle, we need to indicate where the other side goes wrong, not in their conclusion, but in their reasoning, i.e. does A really show that P? Does B really show that not P? And, of course, are there any hidden assumptions in any of these arguments?
 * I feel that the problems with the proposed calculus-based solutions is this: while calculus indeed shows that there is a point in time when we are at the destination point, calculus does not show that we can reach that point in time. To give the example earlier, when we do each operation in half the time as the previous one, then if we do the first operation in 1 second, then after 2 seconds we'll have done infinitely many things. Sure, so far so good: we all agree. But, is the 2 second point in time a point that can ever be reached? Now this seems a strange question: we know that time can flow from now to 2 seconds from now. OK, sure, but here's the thing: the argument that when time has reached the 2 second point we'll have done an infinite amount of things/passed an infinite amount of points, implicitly assumes that time is dense, for we associated the infinite number of tasks/operations/events with an infinite number of points in time between now and 2 seconds from now. But what if time is not dense? Then time can indeed reach the 2 second point, but no infinite number of events will have taken place. Indeed, it is because of this that I charge calculus-based solutions to implicitly employ circular reasoning. Moreover, I also feel that Zeno's argument shows that time in fact cannot be dense: if it was, time could not flow from now to 2 seconds from now (or to any point other than now). By the fact that it does, modus tollens, time is not dense. —Preceding unsigned comment added by 67.248.254.112 (talk) 13:27, 29 October 2009 (UTC)
 * Ah, an erudite thinker joins in at last. :)
 * Good to see that you are all back. I'll reply to a few of User:67.248.254.112 remarks, the most erudite of us all. First the statement: as you go through the infinite sequence one by one, how can you ever reach the end. As I said before, if you have an infinite sequence, then it has by definition no end. And hence, if you go through it one by one, you will never reach the end of the list of steps. However, the above description - go through the infinite sequence one by one - uses implicitly a discrete time model.
 * Furthermore, it is true the sequence of times will never reach 2 in the dense time model, even though it converges to it. Every mathematician will be careful to point that out. And that is not a surprise, because the whole sequence of points contains no point 2. However, as I said before, it is rather unfair to define a series of points of time, none of which is 2, and once all them are completed complain why time 2 was never reached. If you would have defined a series, finite or infinite, that contained 2, you would have had a reason for complaining.
 * So, the central question really is why you shouldn't be able to complete an infinite number of steps. In the dense time model there is no reason why you shouldn't. And it seems we agree on that.
 * And if time and space are discrete there isn't a problem either, because you won't have an infinite series of tasks to begin with.
 * This leave the case where time is discrete, and space dense. In this model we see that in each step, Achilles passes a smaller and smaller distance. This simply means that as time progresses he slows down, to a stand still. However, if Achilles comes to a standstill before he reaches the tortoise, then it's not exactly surprising that he won't reach the tortoise.
 * I have no problems to accept that Zeno's point is to show that a dense time model is unrealistic or counter-intuitive. It might very well be unrealistic, physics will tell, and it is definitely counter-intuitive. But from the mathematics point of view there is no problem. Ansgarf (talk) 01:54, 30 October 2009 (UTC)

I would have to say that I find reading the various responses on this page quite entertaining. To watch how some attempt to ignore, or deny everyday aspects of reality ... fascinating.

But Ansgarf, pray tell, what exactly is your problem with accepting the central paradox of life? Why do you have apparent difficulty understanding and accepting the "inseparable-duality" of individual and community, of wave and particle, of local (Relativity) and nonlocal (at-once, quantum) connections?

"Mystics have spoken to us through the ages in terms of paradox. Is it possible that we are beginning to see a meeting ground between science and religion? When we are able to say that a human being is both mortal and eternal at the same time, and light is both a wave and a particle at the same time, we have begun to speak the same language." — M. Scott Peck

As explained in the above section "Clarity of cause" I can appreciate your wanting to avoid responsibility for the reality you experience. I get that. But it need not be feared to the extent that your denials give voice. We're all in this together; you're co-creating your lived experience within the rich possibilities of the underlying multiverse. Arguing that mathematics (based on continuity/calculus/infinite-series) solves Zeno's paradoxes contrary to consistent (quantum) physical evidence (that physical movement is NOT fundamentally continuous) telegraphs a disconnect of extraordinary proportion, and does you a disservice.

In particular, your "Zeno's paradox assumes indeed that time and space are dense" is incorrect. Many of the proposed solutions are based on that assumption, correct. But the resolution to the paradox of motion need not be reliant on that assumption. In any case, assuming that time and space is dense does not resolve the paradox. I might as well say, "time and space is made of fairy stuff and all those angels on pinheads make it all happen" and therefore the paradox is solved. You offer no grounds or ideas as to why dense space-time is able to resolve Zeno's Paradoxes.

In any event, if your world-view was congruent with observed reality, you would be able to resolve many of the dilemmas facing quantum physicists (e.g. string theory) as to how particles (and thus conglomerations of particles - such as an arrow, runner, or whatever) move through the quantum scaled, and sub-quantum scaled increments - which, as has been clarified many times before, is a necessary and unavoidable requisite if one accepts infinite series solutions (as all such solutions require infinite, unending, endless, seamless continuity and contiguity of PHYSICAL steps in movement-- contrary to the observable, and repeatable evidence of quantum physics).

Might I recommend that you make like a good scientist by beginning with the evidence (of quantum theory, everyday life) and working a theory that fits the world you know, rather than one you wish were true?

I see that, despite work commitments requiring my attention, I'll have to put some time into setting up formal mediation/arbitration to begin more disciplined reworking to remove the clear bias and unsupported assumptions in the main article -- so who's up for being the other respondent? JimWae? Ansgarf? or both of you (one, two, or ten thousand: same to me) Steaphen (talk) 09:19, 29 October 2009 (UTC)

btw, Ansgarf, re your "Given that there are people who can't resist to bring up QM, we should keep mentioning it. But mention it as having little relevance for the paradox." Really? Please explain in mathematical and conceptual detail exactly what is happening when an arrow (in particular the lead atom in an arrow) passes through sub-Planck scaled physical increments in movement, which (by the requirements of infinite series) it MUST do at some point (many points). Do you not understand the central and pivotal role that QM must play in the issue of Zeno's Paradoxes, by simple virtue of the fact that all physical things MUST enter into, and pass through scales of incremental movement where QM reigns supreme to all other sciences? What exactly are you suggesting occurs at such small increments? What experimental evidence do you have to support your ideas?


 * Perhaps the next person to respond on this page should get to be the other respondent in formal mediation. Steaphen (talk) 19:43, 29 October 2009 (UTC)


 * BOO! Nowhere does the article say "infinite amount of time", and and it has not since this 2008-SEP-12 edit, having been tagged here on 2008-MAY-22. That was, however, Aristotle's argument vs Zeno. --JimWae (talk) 22:02, 29 October 2009 (UTC)
 * Ah, Jim, thank you for volunteering. Soon. Steaphen (talk) 00:01, 30 October 2009 (UTC)


 * Oh yes, the article did use the "taking an infinite amount of time" a while ago. I remember someone was talking about Homer running for the bus and using that phrase. I changed it before the whole example was taken out completely. But it should be pointed out that many, if not most, descriptions of Zeno's paradox on mainstream webpages use this phrase, or at least in their 'rebuttal' end up saying "... and so you *can* do an infinite number of things in finite amount of time".  —Preceding unsigned comment added by 128.113.89.96 (talk) 01:33, 30 October 2009 (UTC)


 * Here is what the article says under 'Proposed Solutions': "Using simple algebra, we can calculate the distance and time at which Achilles would match the position of the tortoise: 111 1/9 metres after running for 11 1/9 seconds. This is neither an infinite distance, nor an infinite time." The last line here is clearly meant as a rebuttal to the claim that it takes an infinite amount of time to pass the tortoise. But that claim is part of the straw man version of Zeno's argument as explained above. This is therefore not a solution to the paradox. Interestingly, it continues by saying: "While this solves the mathematics for this one paradox, ...". Well, I suppose that's right: if the question is "when and where does Achilles pass the Tortoise", then indeed we can use calculus to give a pretty good estimation (of course, if space and time are not dense, it may be off by some really small amount, but good enough for most engineering purposes, sure). But, that's not the same as solving the paradox, because in Zeno's paradox the crucial question is not about *where* or *when*, but about *how*: "How is it metaphysically possible for Achilles to pass the tortoise?". Calculus provides no answer to that question, and thus does not resolve the paradox. —Preceding unsigned comment added by 128.113.89.96 (talk) 20:49, 3 December 2009 (UTC)

A few answers for Steaphen. But first a remark to the others who are reading along, and wonder what Stephan is talking about. We two are involved in a discussion in a different forum, and most of his comments referred to that discussion. I'll ignore that part of his reply here, and address it in the appropriate forum instead.

Steaphen, you made a comment about what good scientists ought to do. I won't comment right now, but just point out, that the definition of "dense time" doesn't mention the stuff it is made of, but lists a number of mathematical properties. And none of them involve "fairy stuff". If you ridicule proper formalisations, it creates the impression that you simply aren't aware of them. So, as a good scientist, you first should look up the definitions.

Steaphen, QM affects indeed all of physics. But Zeno's paradox is a thought experiment. He assumes, for example, that in between any two points in space there exists another point. This might not be true in reality, but is still an assumption of Zeno. It is possible to imagine points in time and space, even though they might not exist in reality. We are not bound in thought experiments (or mathematics) to the physically possible. In QM the wave represents the probability distribution in space/time. This means that one of the core assumptions of Zeno's paradox, that a particle is at a certain point in time doesn't hold. If then add the claim that time/space isn't dense to begin with, you also killed a second assumption of Zeno, namely that in-between any two points there exists another. Ansgarf (talk) 01:52, 30 October 2009 (UTC)
 * BTW: I see no reason for mediation. I don't disagree with most of Jim's or User:67.248.254.112 observations. If anything the discussion has clarified the assumptions Zeno made or didn't make.Ansgarf (talk) 01:52, 30 October 2009 (UTC)


 * Dear Ansgar,
 * re your "But Zeno's paradox is a thought experiment." and "it is possible to imagine ..." many things. It seems to me that your thought experiments, by decoupling them from direct application to physical reality, can be of any nature, on any subject, with equal validity - hence my reference to angels on pinheads. Believe it or not, how many angels might fit on pinheads was once considered a serious question. So, what are you using to keep in check your speculations, if not the backstop called "reality"?


 * As far as what Zeno assumed, we are in no position to assume what Zeno considered or believed. In any event, it is irrelevant what Zeno thought. The central issue is of explaining in detail how things actually move. If mathematics can shed light on that phenomenon, (verified in experimental fact) then well and good. But I have not seen any experimental verification of perfectly continuous and contiguous physical movement (upon which all infinite-series solutions are reliant). As far as I'm concerned, any discourse that does not reference the experimental physical evidence, has scampered off into cloud cuckoo land, along with those fairies.


 * As for the justification for mediation (more likely arbitration), I think it is time that the nonsense surrounding the widely- accepted solutions to Zeno's Paradoxes (involving infinite-series) is shown to be what it is ... an unsupportable silliness without basis in fact. Many have considered religious folk of times past (in Galileo's time) to have behaved poorly, and to have held up the advance of science. The dogmas and blind superstitions that are endemic in the discourse on Zeno's Paradoxes rank far worse in my opinion, for at least with a religious world-view, we have (maybe, sometimes, possibly a little bit) a benevolent entity who offers salvation if you're nice enough. But with (contemporary) "scientific" views, we are hapless dorks in an impersonal universe. And atheists wonder why religions are on the rise.


 * As The Belief Doctor, it's my job irrespective of crowd opinion to the contrary, to be a ghost .. sorry, "dogma buster", be they scientific, religious, or new-age. Steaphen (talk) 07:41, 30 October 2009 (UTC)


 * btw, Ansgar, in having met you, I can attest that your atheistic "black or white" world-views pale in comparison to mine at your age. I empathise and applaud your enthusiasm for your ideals, but perhaps unlike me (of stubbornly remaining closed to possibilities, and my intuitions), you can take advantage of the latest advances in quantum theory to advance understanding into the deeper nature of reality, at a relatively young age. As I've explained on Richard Dawkins website, atheists are good folk seeking to better the world, but too often "throw the baby out with the bathwater" - to the detriment of one and all. Kind regards, Steaphen (talk) 07:58, 30 October 2009 (UTC)


 * 1. Zeno's argument was of a mathematical nature, and hence it is fair to use mathematics to discuss it. You apparently think that just because mathematics deals with entities that do not have a physical reality you can make it up at will. That is however not quite true. Your reference to "angels on pinheads" suggest that you might not be aware that a mathematical proof is actually fairly rigorous, and not a fantasy. Even if none of them involves physical experiments. The fact that infinite series in dense time can converge is independent from physical reality. However the mathematical fact that you cannot divide a finite distance into an infinite number of distances with a minimal non-zero length, does mean that you cannot do this in reality either. No matter what your experiment tells you. Mathematics restricts physical reality, not the other way round.


 * Of course, whether a certain mathematical assumption can be applied to physical reality does depend on physical reality. And if in reality time and space is not dense, it simply means that you cannot apply a theory based on dense time/space. And with respect to Zeno it simply means that if space is not dense, you cannot divide a finite distance into an infinite number of smaller non-zero distances of minimal length. So, if you are right that space and time is discontinuous and not even dense - that there is no and never will be experimental verification of perfectly continuous and contiguous physical movement - then the whole problem is solved in an instant.Ansgarf (talk) 12:23, 30 October 2009 (UTC)
 * 2.A response to your other arguments that were mostly of a personal nature. I am not quite sure, you might mistake me for some other person. You can impossibly know much about my beliefs from our brief encounter, given that we were only discussing cardinal numbers. Whatever you think my beliefs are, they are either someone else's, or, as likely, projections.


 * Also, you might be a bit mistaken about my age; I apparently look a lot younger than I actually am. Concerning the views that you apparently held not too long ago when you were my age. They are entirely your responsibility, and not my business. Good on you if you were intolerant and bettered yourself. Finally, mediation. I doubt there is need for mediation with any of the other participants; despite the differences the discussion is civil, and mostly on topic, and we are not even involved in an edit war. I know that you were involved in mediation recently, but I think we are able to sort this out the usual way.Ansgarf (talk) 12:23, 30 October 2009 (UTC)

You


 * Dear Ansgar,


 * The impetus for mediation (-> arbitration) is to engage more disciplined treatment of this subject. Your responses are examples of that need -- nowhere that I can see have you attempted to correlate (mathematical) theory with experienced (quantum) reality. The process of arbitration should bring some focus onto the bias and unsupported assumptions in the main article, and the lack of validity of using infinite-series when dealing with Zeno's Paradoxes. Cheers, Steaphen (talk) 03:09, 31 October 2009 (UTC)
 * Maybe you are right and I didn't explain myself sufficiently clear. First, I made a few references to QM that I am happy to clarify. When I mentioned "minimal distance" I meant something like the "Planck length", when I referred to discrete time, I meant a model that has a minimal time, like the "Planck time", when I referred to the probability distribution in space/time, I was referring to the Copenhagen interpretation of the solution of the Schroedinger equation. Maybe I should have stated this more clearly, because it seems you didn't get the references. My apologies.
 * Also, it very well be possible that my treatment isn't as rigorous as it should be. But I am gladly repeat the sturcture of the argument. What I tried to convey is they following case distinction:
 * If time and space is dense the calculus solution applies.
 * If time is discrete and space dense, it means that Achilles passes smaller and smaller distances with each step, i.e. he stops before he reaches the tortoise.
 * If space is discrete (i.e. there exists a minimal length) then it is impossible to divide the distance into an infinite number of non-zero distances.
 * I hope the structure got clearer.
 * If the latter case corresponds to reality - supposed space/time is not dense - then there is no need for a calculus solution. If you attack the calculus solution because you believe space and time is not dense, you are actually barking up the wrong tree. Ansgarf (talk) 08:37, 31 October 2009 (UTC)


 * btw, Ansgar, your "Mathematics restricts physical reality, not the other way round." is quite an extraordinary statement. I'm not sure you meant to say it as said. There would be many on the planet today who wouldn't give a rat's arse about mathematics .. they're happily getting on with their physical reality, expanding it, enjoying it, creating it despite your protestations, or calculations to the contrary. Mathematics is only relevant if it is useful for people to better create their reality. Otherwise, we can entirely dismiss it, discard it, or "Centwurion, throw it to the floor" (despite what the wabble want! :) Steaphen (talk) 07:56, 31 October 2009 (UTC)
 * It is quite an extraordinary statement, isn't it. And I actually meant it exactly as I said it. The number of people who care about mathematics doesn't really matter to mathematics rule over physics. To paraphrase an example: Suppose there is one person who won't give a rat's arse about mathematics meeting another person who won't give a rat's arse about mathematics in an empty pub, then there will be two people who won't give a rat's arse about mathematics in the pub. No matter how big their disgust to mathematics is, how drunk they are, how illiterate, they won't be three, not 1.5, not Pi, but exactly two people who won't give a rat's arse about mathematics. Even they adhere to the laws of addition for the natural numbers, whether they chose to or not. Ansgarf (talk) 08:46, 31 October 2009 (UTC)


 * Sorry Ansgar, but here again you are "not entirely correct." In which probable reality are you claiming your statements to be true? In the superpositions of possibilities, I grant you some may adhere to your concepts about reality, but others might not (in which realities the constants of that physical reality differ from ours). You are in no position to assert otherwise (well, some of your alternate selves may comment appropriately, but not "you"). This again reflects your either-or thinking. "This reality or nothing," you say. But, here's the thing ... the multiverse takes all kinds!, so you're okay, donta worry, well, not too much :) Cheers, Steaphen (talk) 09:05, 31 October 2009 (UTC)


 * Methinks you confuse descriptions of systems (e.g. mathematics), with the systems themselves. A bit like confusing the menu with the food. Running late, now. Will pop back in a few weeks. Ciao.
 * The case distinction was mentioning different models of time fairly explicitly, and it clearly distinguished them from reality. You must have overlooked it, since you were in a hurry. A multiverse is a very interesting idea by itself, but doesn't really address your problem. In any multiverse the basic rules of addition apply to anything that can be reasonably counted by natural numbers.  Also, you were talking people on this planet not in possible multiverses; bringing up a multiverse looks like a smokescreen, to conceal that for this universe you have no retort.
 * Ironically, the multi-verse model is a purely mathematical model. No one has been able to experimentally prove it as far as I know. That is fine with me, I find it intriguing nevertheless, but I'm afraid it doesn't satisfy your own standard of experimental validation of mathematical models.
 * Talking about experiments. If you think that my prediction about the two people who won't give a a rat's arse about mathematics isn't true, I am eagerly awaiting your experimental proof of the opposite. When you return. Hope the food is fine. Ansgarf (talk) 13:40, 31 October 2009 (UTC)

Have you read the article lately? I could never figure out what your real objection was to any specific part of the article - unless you are wanting to remove all discussion of infinite series from the article, or have the article make some authoritative claim about what space & time "really are", and what proposed solutions are totally right & totally wrong JimWae (talk) 03:45, 31 October 2009 (UTC)


 * Hi Jim, The reason for what I expect will be the necessity of arbitration is your intransigence over the idea that we can mathematically determine when the runner will overtake the tortoise. Specifically "Using ordinary mathematics we can arrive at a specific time when and place where Achilles would be able to catch up to the tortoise" This is just plain and simply wrong. Period. This is the disconnect that needs to be addressed.


 * We are not justified in assuming that with mathematics we can do any such thing, IF that calculation has NO correlation with fact, anymore than calculating angels on pinheads has relevance to the subject at hand.


 * Do you not understand that simple expedient of the scientific method; of correlating theory with evidence? This continued and deleterious obstinacy in the scientific community is a travesty of modern science, and needs to be corrected for all our sakes, including mine! (This little freddie ain't gonna sit by while the water boils! :)
 * Cheers, Steaphen (talk) 07:23, 31 October 2009 (UTC)

If Achilles runs ten miles per hour, and the tortoise runs one mile per hour, and the tortoise has a two mile head start, then the distance Achilles runs is 10 t miles and the distance the tortoise runs is 2 + t miles. Setting those equal, 10t = 2 + t. we find that Achilles passes the tortoise after 2/9 hours. Now, we do an experiment, with two runners with the given speeds and the given head start. In the experiment we discover that, within the margin of error of our measuring instruments, the fast runner overtakes the slow runner after 2/9 hours. The experiment follows the mathematical model.

Incidentally, a friend of mine has proved that exactly 2.9 x 10 ^ 23 angels can dance on the head of a pin, if and only if it is first established that angels can dance.

Rick Norwood (talk) 20:07, 31 October 2009 (UTC)


 * 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 --JimWae (talk) 21:00, 31 October 2009 (UTC)
 * The issue with angels on the head of a pin was not to find THE SUM, but whether there was ANY limitation - did such "entities" have extension & occupy space. Let's drop the pins & let the angels do whatever --JimWae (talk) 21:05, 31 October 2009 (UTC)

But Zeno, in his paradox, was not anticipating quantum mechanics. He was trying to illustrate that large scale motion was an illusion. The paradoxes of quantum mechanics are not the paradoxes of Zeno. There is a tendency to attribute to ancient writers ideas that could not possibly have been part of their worldview.

The issue with angels was, from the very beginning, a joke, making fun of the Thomists.

Rick Norwood (talk) 12:08, 1 November 2009 (UTC)

The paradox and modern science
In the previous discussion I already tried to address that in my opinion the mentioning of QM is somewhat confusing and sophomoric. Unfortunately we got sidetracked. Anyway, in my opinion the problem remains, namely that a few modern theories are mentioned in the entry, while their relation to the paradox is unclear at best. Mentioning these theories is simply some form of name dropping.


 * The section of QM needs to be rewritten. First, Zeno assumed that you can divide distances infinitely often. According to QM this might be wrong in practice. If anything, it makes the whole formulation of the paradox impossible for quantum systems. The only aspect on which Zeno might be right is that time cannot be divided infinitely often, and thus that infinitely many "nows" do take an infinite amount of time. But that is only half of the paradox.
 * The current entry suggests that it is somehow important that you cannot measure small distances. Nowhere in the paradox is something measured, so this remark is at best confusing.
 * I looked for references about Brouwer and Zeno's paradox, and all I could find is a remark that Brouwer rejected a dense time model. Intuitionist do reject the law of the excluded middle on infinite domains; this does however not mean that they deny that infinite sums can converge. There exist intuitive formulations of calculus that deal with this fairly well. That said, intuitionist might reject the notion that you can divide time and space infinitely often, i.e. which would undermine the whole formulation of the paradox. And to mention Kronecker is in my opinion just sophomoric. Somebody apparently remembered his name in conjunction with constructivism, but that is not enough to include him here.   What can be said about the paradox and constructivists is that Zeno's problem's with the notion of infinitives are echoed by modern day constructivists. But they do not much more than echo them Ansgarf (talk) 05:52, 3 November 2009 (UTC)

There is one area where Zeno's name is used in modern science, and that is real-time design and verification. When you design or verify such a system, you want to exclude so-called Zeno behavior, since it is impossible to implement such systems. I might add that to the article. Ansgarf (talk) 05:52, 3 November 2009 (UTC)


 * Dear me. Ansgar, the act of observation is a form of measurement. Your choices (of what to focus on, observe, and when) in effect take measure, create it, discern it, differentiate (collapse possibility into lived experience). "Nowhere in the paradox is something measured," ... ? this is exceedingly odd thing to say. The infinite-series solutions involve the measure of distance and time, do they not? Zeno observations involve good measure of the circumstances (or was it all vague, fuzzy and cloudy for him)? Also, your "Ironically, the multi-verse model is a purely mathematical model. No one has been able to experimentally prove it as far as I know" ... crickey, what do you think quantum superpositions are? Again, what exactly do think is physically occurring that you think you can explain with infinite-series solutions? And how is that verified by experimental evidence?


 * In brief, the need for arbitration is to, again, bring some discipline to this article. For example, from the section above, to suggest that "orders of magnitude" have relevance when calculus (or infinite-serious solutions) involve and REQUIRE ALL orders of magnitude is an extraordinary disconnect between application of theory to real life, with the theoretical underpinnings.


 * Clearly, the majority of respondents on this thread/page will continue in the same style of ignoring the disconnect between theory and actuality.


 * Just a quick thought (I've already taken too much time here) ... arguing that someone is guilty of being a witch and needs to be burned at the stake is akin to arguing that physical reality is 'guilty' of continuity.


 * Reasonable person: "What's your proof for witchery (continuity)? What evidence supports your suppositions?"
 * Dogmatic/religious person (from hereabouts): "Uhm, well, it's just obvious, ain't it? Besides they look (it looks) guilty of witchcraft (continuity). And what's more, the good book says so. Look, it's written!. The sacred texts say so (the Gospels of Science). So let's just, well, heck, burn 'em. (ideas and evidence suggesting otherwise)"Steaphen (talk) 03:46, 6 November 2009 (UTC)

Methinks that maybe it wasn't only you priests from Galileo's time who reincarnated as modern-day atheists/sceptics/scientists, but also a sizeable section of the rabble who saw off Giordano_Bruno.

Hi Stephean I answer your comments in order.
 * "The infinite-series solutions involve the measure of distance and time, do they not?" No, they don't. Zeno assumes that Achilles and the tortoise have a well defined position, but not that the position is measured.
 * "What do you think quantum superpositions are?" They are a mathematical description of quantum effects. To describe them complex numbers are used, i.e. numbers with a real part and an imaginary part.
 * On "orders of magnitude". They are not mentioned in the article, only on this discussion page. Since the paradox doesn't measure anything, the order of magnitudes is indeed somewhat irrelevant for the calculus solution. But that was Jim's point, so you ask him what he meant; I usually value his opinions and contributions.
 * Do you want arbitration with us? Or do you want to reopen the previous case of arbitration that involved you and a few others?
 * So what if physical reality would not be "guilty" of continuity? It would be fantastic. Because in that case there is no infinite series of distances, and thus no paradox.
 * Your insistence on experimental evidence trumps mathematics and logic is ironic in connection with Zeno's paradox. This because, Zeno's argument is that the the experimental observation of motion must be an illusion, because it leads to a logic contradiction. So, Zeno uses explicitly the assumption that what is in conflict with logic cannot be reality. Exactly the opposite way you claim to argue.
 * But lets stick with your paradigm that in the end experimental evidence counts, in the tradition of Galileo. Can you provide experimental evidence that you have for the claim that a runner cannot pass a tortoise? How many runners do you need, how many tortoises, and what is the experimental setup? Ansgarf (talk) 12:38, 6 November 2009 (UTC)


 * Ansgar, seriously, this is all entertaining up to a point, but ...


 * You are most welcome to misquote me, or misinterpret my work, but the point I've made consistently throughout my posts, is the validity and efficacy of combining theory with evidence, or if you like, of combining evidence with theory. That is not in any way discounting the validity or necessity of theory (I enjoy the technologies of modern life that have been derived from solid theoretical foundations). I have not suggested or implied that experimental evidence "trumps" theory, any more than an individual "trumps" the community (of which he/she is part). It would seem you've entirely missed the meaning of the Table of One AND All.


 * I have not suggested that a runner cannot surpass a tortoise. That would be silly. What I have questioned (and continue to do so) is the explanation as to how the runner overtakes the tortoise.


 * I include the experimental evidence of the world's "most successful physical theory in history" in my conceptual frameworks. I don't make excuses why they don't apply.


 * If you wish to discount or ignore aspects of reality (quantum experimental evidence) when attempting to explain how things operate and move in the quantum domain (which AGAIN, they must enter, due to the infinite-contiguity required by infinite series) then you are most welcome to do so, but I will call you and others to account for such glaring disconnects between reality and theories purporting to explain said realities.


 * The arbitration is inevitable because of the style of responses that you and others adhere too. Those responses are simply unsupportable superstitions with no basis in experimental fact. The onus is not upon me to prove anything. It is on those who assert that "Using ordinary mathematics we can arrive at a specific time when and place where Achilles would be able to catch up to the tortoise." Quantum theory and evidence shows this is nonsense. Plain and simple. It can't be done.


 * As for quantum superpositions, I asked, "what do you think is physically occurring". Your avoidance of offering any suggestions as to what is physically occurring telegraphs your disconnect of theory with fact. That's "witch-burning territory" which fuels the rank, first-order superstitions that we see gaining ground in the world today (scientific and religious fundamentalisms). They belong to another age. Not here, not now. Steaphen (talk) 00:01, 7 November 2009 (UTC)


 * Hi Stephaen,
 * You might have argued consistently that physical evidence and theory go together, but many of your statements seem to say that a mathematical theory is plain wrong, because there is a case (quantum systems) where it cannot be applied. Look for example at this statement:


 * If "Brand A mathematics" (calculus) cannot be used in the detail of explanation (as the evidence of quantum physics now reveals), then get rid of it..


 * This is not the only one -you might recall your statement about people who give a rats arse about mathematics - but to be fair, maybe you didn't mean it as I understood it. So to clarify.  Are you claiming that  (A) you cannot apply calculus to a quantum system, because time is not dense, or are you claiming (B) the calculus solution is in itself wrong, because you cannot apply it to quantum systems. Or do you claim something else?


 * Sorry, if I didn't answer your question about quantum superposition to you satisfaction, but you asked two seemingly disconnected questions. It wasn't quite clear that "physically possible" referred to "quantum suppositions". As far as I know there are at least two interpretations for superpositions, one is the Copenhagen interpretation, and the other the many-world theory. One says that they describe a probability distribution, the other assumes that all possibilities are realised in multiple worlds. However, each of these worlds has the same  fundamental laws as ours, and you should expect there the same to happen under similar circumstances as here. But as said, many worlds is just one interpretation, CHI is another. And it is still a mathematical model that explains measurements fairly well.


 * If you want to know what infinite-series solutions can explain at all, they can explain a lot. For example, when you deal with probability distributions. Or think of differential equations and integrals. Or of algorithms to compute square roots. There are plenty of infinite-serries solutions that we use with great success, in engineering, physics, chemistry, biology.


 * You said that the point at which Achilles passes the tortoise cannot be calculated. This is a universal claim, and all it needs is a counter-example that you asked for. I hope you take an analogue clock as substitute. My kitchen clock has a hand for seconds which moves 60 times faster that the hand for the minutes. The infinite series solution tells me that if the hands for minutes and seconds align, it will take 61 second before they align again. Given some uncertainty in me measuring the time. According to your claim this cannot be the solution. But I went, and found that after 61 seconds the second hand had passed the minute hand. The same happened when I checked a few minutes later. So, I would conjecture that you can use the infinite series solution. I would even go so far to claim that you can use it for any two objects (of moderate size), one trailing but moving faster than the other, both at constant (non-relativistic) speeds. So, the ball is now in your court to demonstrate the opposite, given my universal claim about moving objects.


 * Your table-of-all doesn't really say anything useful about this topic. It is a collection of stereotypical pairs (and never triples), some of them are set-subset pairs, others antonyms, others just near antonyms, others complements, and you claim that everything is separate AND one, so that in the end it is fairly arbitrary. No matter what you want to say, you can probably find a pair in the table and come up with some analogy that fits the case. Using the table I could easily argue that theory is unlimited, while experiments are limited. And it would be kind of true as well.


 * I commend you for your attempt to use the "most successful physical theory in history" in you conceptual frameworks. But I am not convinced that you do. Granted, the table-of-all is some kind of meditation on the concept of duality, present in quantum mechanics, but that doesn't really qualify as using it. It is suspicious that you call it the "most successful physical theory in history" while you question that infinite-series solutions can explain anything physical. This because the Schrödinger equation is a partial differential equation, the wave function is an integral, and the possible states are defined as complex numbers. These involve by their very definition infinite series solutions, and this is, by your own admission, the "most successful physical theory in history".


 * To finalise, if you ask me whether I understood the table-of-all, it is fair to ask if you understood the case distinction I gave. Because it covers the case that space/time is not dense. And in that case you don't need to apply the calculus solution, even if it would be in many cases a decent approximation. In that case there is no need for an infinite series solution, because in that case there is also no infinite series. Ansgarf (talk) 14:08, 7 November 2009 (UTC)


 * Ansgar, please be more diligent. I asked what is physically occurring, not about mathematical models that have no evidential relevance to reality. Irrespective of however much calculus is used in quantum wave-functions, there is NO, and I repeat absolutely NO calculus that can be applied to the actual mapping or exact prediction of particle behaviour. NONE. It's called the 'collapse of the wave-function' and there is no known mathematical model to explain it, or to map the process or path of a particle. NONE. Do you get that.


 * If you want to talk about the nature of actual physical movement (of Zeno's arrows, runners and hares), as opposed to your superstitions as to what you wish is occurring, be my guest, and be welcome to the Nobel Prize that will ensue. In the meantime, you and other respondents on this forum need to cease ann desist with the infinite-series superstitions in regards to Zeno's Paradoxes. They are not able to be substantiated, or verified in fact, and thus have no place in this article, other than in reference to how some believed they were relevant to the situation, but are now seen as approximations that do not in fact explain the evidence.


 * Arguing that infinite-series (and Newtonian mechanics) explain movement is directly analogous to arguing the Earth is flat. In any limited, local area context (e.g. one's lounge room) the 'earth' (floor) may indeed be flat (if the builder was professional and diligent), but within an expanded context, is emphatically incorrect. In a limited context, infinite series work well enough. Within the expanded context of quantum-systems, quantum superpositioning (experimentally confirmed for molecules) etc, infinite-series (Newtonian mechanics) are emphatically unable to be used to explain the facts of physical movement.


 * Continuing the analogy, infinite-series is 'approximately correct" for explaining (limited-context/approximate) large-body physical movements, but not for the expanded context of actual, exact large-body physical movement. This article (Zeno's Paradoxes) is about the latter ... the exact, actual nature of large-body physical movement, which is not addressed by, or adequately explained by infinite-series solutions.Steaphen (talk) 20:57, 7 November 2009 (UTC)
 * Nobody here insists on using a infinite-series solution when it is not appropriate. Nobody. It seems that you have one single argument, namely that you cannot use a infinite series solution, because space is not dense. And nobody says otherwise. You are unable to acknowledge the arguments of others, because this is your only argument. Your only argument, an argument that is not relevant at all. Repeating it won't help.Ansgarf (talk) 00:51, 8 November 2009 (UTC)
 * A few words about wave functions. First, by its very nature it the wave function will not give you an exact prediction. If you mean by exact a deterministic prediction of measurements. This because the wave function can be understood as a probability distribution. The evolution of the wave function is described by a differential equation that we know as Schroedinger equation. NO calculus solution? The wave function is a calculus solution.
 * Also the statement that there exists no mathematical model to explain the 'collapse of the wave-function' is very odd, knowing that the 'collapse' is the name for a reduction of the eigenbasis of the wave-function due to a mathematical operation modelling measurement. In short, the 'collapse' is a mathematical model. Ansgarf (talk) 12:36, 9 November 2009 (UTC)

re your "One says that they describe a probability distribution, the other assumes that all possibilities are realised in multiple worlds" -- probability distribution of? ... what? Fairies, angels, vampires, cloud-cuckoo land inhabitants? Steaphen (talk) 20:27, 7 November 2009 (UTC)


 * Is is a probability distribution over possible states. And all physicists will tell you that this is a very successful mathematical model, and to ask what actually happens physically below this level is pointless. It just shows that you hope to expect a mechanical world down there. Your question tells us that you do actually not believe your own claim, namely that space is not dense. Otherwise you would know that below a certain threshold, there is no point to ask what happens.  Ansgarf (talk) 00:51, 8 November 2009 (UTC)
 * To make some headway I'll ask you again:
 * Are you claiming that (A) you cannot apply calculus to a quantum system, because time is not dense, or are you claiming (B) the calculus solution is in itself wrong, because you cannot apply it to quantum systems?
 * Did you notice that we covered the case that time and space is not dense. And that nobody claims that in that case calculus is the solution?
 * Ansgarf (talk) 01:08, 8 November 2009 (UTC)


 * "All physicists". Incorrect. I can cite many examples to the contrary.


 * "there is no point to ask what happens." You have got to be joking. Seriously, you can't really be suggesting that you can offer mathematical models while asserting that there is no point asking what actually happens. The subject of Zeno's Paradoxes is exactly about what happens physically -- its about explaining how runners, arrows and other physical, everyday objects move.


 * As your responses have degenerated into a level of nonsense that beggars belief, grounds have now been well-established for actioning formal mediation/arbitration. Steaphen (talk) 01:12, 8 November 2009 (UTC)
 * I am actually suggesting that on quantum level and below all we have are mathematical models, that reasonably well predict what happens. And that is it. I admit that you might find a physicists who thinks he can can explain exactly what happens below that level. So I should have said that it is commonly accepted that quantum mechanics is a mathematical model. And I am also suggesting, that if space/time is not dense it it pointless to ask what happens in between points at minimal distance. If you find that reason for mediation, call a mediator.
 * BTW: Zeno's paradox is not about what actually happens. Zeno asserts that in between any two points there is a point that must be crossed. He says nothing about how you get from point to point.Ansgarf (talk) 02:13, 8 November 2009 (UTC)
 * You didn't answer my questions. Could you please do so. It would help to know your position, especially if you call for mediation.Ansgarf (talk) 02:13, 8 November 2009 (UTC)

A simple question
For all those respondents who argue for infinite-series solutions to Zeno's Paradoxes, I have a simple question.

Consider a runner who is to run a race (or catch a tortoise).

The runner is on the start line, at position 0.000000 metres.

The question is this: What exactly happens when the runner (and every part thereof, including all the atoms in his/her body) begins moving and moves to say, position 0.01 x 10-50 metres?

(As a corollary), What mathematical model can you use that can be directly and unambiguously verified by experimental evidence to confirm the efficacy and relevance of said theory?

For those who might wish to jump the gun here, quantum evidence reveals that at those distances (which the runner MUST pass through, obviously!), the position and speed of the atoms and the conglomeration of atoms we know as "runner" cannot be precisely defined mathematically.

So, what theory exactly explains what happens and can be verified by evidence?

Steaphen (talk) 23:12, 7 November 2009 (UTC) The Belief Doctor

healing and improving 'bodies of belief' - science, religion, politics, health, business ... life


 * You are asking what happens to a particle on quantum level and what mathematical theory can explain it. You are asking for a theory that has been experimentally verified. It's quantum mechanics. You called it the "most successful physical theory in history". I would be happy to explain it in term of probability distributions over states. But you should know it, since you are by you own admission familiar with quantum mechanics. Ansgarf (talk) 01:02, 8 November 2009 (UTC)


 * Dear Ansgar, thank you. See above re now sufficient, ample grounds for actioning formal mediation/arbitration.Steaphen (talk) 01:18, 8 November 2009 (UTC) btw, I was quoting a well-known competent physicist who described quantum physics as the "most successful physical theory in history" (Dr David Deutsch, of Oxford). Others have made similar claims, simply because of the "enormous experimental success" of the theory, unparalleled by any other science.Steaphen (talk) 09:38, 10 November 2009 (UTC)
 * I asked you if you really want me to explain it. But before I do it, I'd really like to know if you accept that quantum mechanics describes satisfactory what happens on quantum scale or not. I would at least know what we debate. Because at this point I am not even sure if you are accepting quantum mechanics as explanation or not. Also with an eye on mediation it would be useful to know. The two questions are:
 * Do you still want an explanation on what happens at quantum level?
 * Would you accept an explanation that refers to quantum mechanics?
 * Ansgarf (talk) 02:21, 8 November 2009 (UTC)


 * The question was "What exactly happens when the runner (and every part thereof, including all the atoms in his/her body) begins moving and moves to say, position 0.01 x 10-50 metres?" and can be verified by experimental evidence sufficient to confirm your theories? It's a rhetorical question. What exactly happens for the runner cannot be defined, or described mathematically regardless of wishful thinking to the contrary. Thus infinite-series are not able to resolve the paradox of how a runner moves. End of argument.


 * Let the Requests_for_mediation/Zeno's_paradoxes (or if necessary arbitration) sort this matter out once and for all. Steaphen (talk) 23:50, 9 November 2009 (UTC)
 * So you didn't want an answer to begin with? Arbitration will only deal with content, so I just want to let you know that I find it odd that you start a new section to ask a question that didn't appear to be rhetorical, and then later you reveal that you were never interested in an answer. The question doesn't seem to be rhetorical and the answer is that the best mathematical model we currently have for these distances is quantum mechanics. But apparently you are not interested in an explanation.
 * You said that the distance is 0.01 x 10-50 metre. You just defined it. Using mathematics. Of course, you wouldn't be able to measure such distances, but that is a different question. Agreed, this is a question that will indeed be addressed in mediation.Ansgarf (talk) 03:56, 10 November 2009 (UTC)


 * A very good point. We can talk about such sub-quantum distances, and such talk does convey meaning to us. However, according to the best theories we have, we will never be able to measure such distances, nor tell what "happens" at that level. Whether space is some kind of "entity" "composed of" some such quantum-distances, we will never be able to claim incontrovertibly. --JimWae (talk) 04:49, 10 November 2009 (UTC)


 * Ansgar, you're way out of your depth here. You appear to assume that I didn't want an answer to the question. That would be incorrect. I was inviting the style of answers provided, as they will provide material for the mediation. This is the quickest way to 'cut to the chase', by highlighting the disconnect in the infinite-series theories, and therefore their inapplicability to the subject of Zeno's Paradoxes. It was and is a rhetorical question. Any competent physicist would tell you that (or anyone versed in the fuller implications of Heisenberg's Uncertainty (Possibility) Principle). JimWae appears to entirely miss the implications of the Uncertainty Principle. It has nothing do with the experimental apparatus. It's a fundamental quality of nature, and simply means that the nonlocal fields (e.g. Bohm's Implicate Order) from which runners, arrows and hares are newly unfolding is and will remain beyond definitive description, or definitive mathematical expression. Infinite-series are applicable to solving Zeno's Paradoxes IF they include superpositions of physical probabilities (but even then they ignore the deeper ground from which possibilities are continually emerging and 'collapsing' into our single-past reality -- an at-once cycling through probabilities that the MWI adherents have yet, it appears, to understand). Infinite-series are categorically not applicable for this linear, single-past reality. That is verified by experimental evidence. If you have any evidence to the contrary, you are both more than welcome to provide it (also rhetorical). Steaphen (talk) 13:17, 10 November 2009 (UTC)
 * Hi Steaphan, maybe I am out of my depth, but I might have misunderstood you because I understand under a 'rhetorical question' a question to which no answer is expected. To call a question rhetorical, and still ask for an answer is odd imho. Furthermore, it is also odd that you asked the question to get material for a mediation. At that point mediation was not initiated yet, which cast doubt on the fact that you initiated it in good faith. Of course, this for the mediator to decide, but it is one arguments that makes the whole process doubtful.
 * So, assuming that you want an answer to the initial question the following. The position of a particle on quantum level can be obtained from the wave function. However the position is not given as a point, but as a probability distribution (under CHI interpretation).  The evolution of this wave function is described by the Schroedinger equation. This is a continuous time partial differential equation.  So, on the small distances that you assume, the particles did already with some probability pass each other, with some probability they didn't. And these probabilities change as time elapses. When you measure the particle the wave function collapses (according to CHI). This is not due to imperfect devices, but as you note, a fundamental property of measurement. And that measurement is subject to the uncertainty principle. And from the point of measurement  the wave function will continue to evolve as defined by the Schroedinger equation, as said a continuous time differential equation. What does this mean physically? I don't know. The math seems to work that good that some call it the "most successful physical theory in history". There are plenty of interpretations of what it means, but regardless, the math, using plenty of calculus, seems to work just fine. Ansgarf (talk) 14:18, 10 November 2009 (UTC)


 * Dear Ansgar, good faith or not, this is about correcting errors in the main article. Don't get too precious. As for the rest, you're only confirming my original point, that probabilities become important, and therefore infinite-series solutions which are based on strict, hard 1:1 correspondences of physical locations with physical objects are wrong. They cannot be meaningfully used to solve Zeno's Paradoxes. Steaphen (talk) 15:00, 10 November 2009 (UTC)
 * It is very difficult for me to discern your original point, even more since you seemed to ridicule probability distributions no too long ago when I mentioned them. Zeno's paradox is defined under the assumption that the position of every object is a point. Therefore, when discussing the paradox it just fair to use the assumption as well. If you want to interpret the paradox in terms probability distributions, you might instead of the distribution use the expected value of the distribution. See also Correspondence principle. The expected value would be a point after all, and it could even be a point that lies in between possible values of the original distribution.Ansgarf (talk) 22:47, 10 November 2009 (UTC)

Dear Ansgar, JimWae et al.

The disconnect between your suppositions and actual physicality, implicit and explicit in your replies beggars belief. If you do not, and cannot "tell what happens" then you have no grounds by which to claim that 'using ordinary mathematics we can calculate ..."

Clearly, that disconnect is blind to both of you (JimWae and Ansgarf). That disconnect is a violation of fundamental scientific-method principles that neither of you seem the least interested in applying. If your theories cannot be substantiated in fact, or congruently applied to the reality we share, then you are no different to any superstitious group, including those of a religious, or new-age nature. In fact you are arguably worse, because science is the main story in our culture, and therefore those who misinform or apply superstitions to such a field are doing a great disservice to humanity.Steaphen (talk) 07:46, 10 November 2009 (UTC)


 * If space were itself discrete, there is no infinite series & there is no paradox of motion. You have not responded to that yet, other than to say you are right & everyone else is wrong. So when does the mediation begin? --JimWae (talk) 09:45, 10 November 2009 (UTC)


 * Dear JimWae, "If" space is discrete then the 'gaps' (the void or ground out of which space unfolds) is infinite (encompassing infinite alternative probabilities). The paradox is even more evident, in that the runner traverses that which cannot be physically traversed, yet does so anyway. Paradox is central to existence, in every manner, none less so than the simple ubiquitous paradox of individual (particle, person, planet) while being, in a deeper sense, the whole (atom, community, universe); of finite while being infinite; of the measurable-particle while being the immeasurable-wave; of part while being whole; of conscious while being unconscious; of ordered while being chaotic; of being different while being one (and the same); and so on, ad infinitum.
 * There are no exceptions to this central paradox of life, at least none you will find or cite short of infinity. And any exceptions you may wish to argue are reliant on disconnects that ride the underlying ground that everywhere and every-when interconnects and interpenetrates. All of which simply reinforces the paradox. Steaphen (talk) 12:53, 10 November 2009 (UTC)
 * Did you just say that in between the discrete points in time there are other points that need to be traversed?Ansgarf (talk) 14:20, 10 November 2009 (UTC)


 * THe short answer to your question is "Yes AND no". The long, more involved answer is "Yes AND no." There are points, probabilities, which remain "non-points" in raw physical terms, but which are real-enough for quantum physicists to build quantum computers, reliant on quantum superpositions. Hence the Yes and no answer. But as far as this physicality is concerned, the answer is an emphatic NO, there are no "points" to traverse, because they aren't physically existent (at least not in this probability). But again, even then "it" doesn't stop there. These non-points (alternate probabilities) are still unfolding from deeper non-local fields of potential.Steaphen (talk) 14:51, 10 November 2009 (UTC)
 * This is a very particular interpretation of quantum mechanics, and it is just one of them. Regardless of whether your interpretation is actually consistent with QM and mathematics. What surprises me is that this position seems to contradict the strong many-world position that you seemed to endorse earlier, since in that model the 'probabilities' are actually existing alternative worlds. Could you once define you position such that we know what to discuss. Ansgarf (talk) 22:47, 10 November 2009 (UTC)


 * Dear Ansgar, Your attempt at putting me in some idealogical box is the problem. I embrace paradox, so while MWI (Many Worlds Interpretation) has some elements of validity, on the whole MWI misses the point. MWI is still a mechanistic model which to the extent that "things' exist (in whichever reality) I concur with, but I go well beyond such limited perspectives.
 * Those probabilities (of which we are speaking) are existing, yes, but also they themselves remain in potential. Apply the paradox of our own "actual-particle within possibility-wave" reality across the board, and you will begin to understand my views. In other words, any probable future has its own probable future, past and present. So they're both existing, while not, concrete but fluid, possible and actual. Once you begin applying that fundamental paradox, you can't go wrong. You'll find no exceptions to the validity of that model. None. You're welcome to try, though.
 * Applying that paradox (of actual while being fluid-possible) to Zeno's Paradoxes "solves" them in that in each step along the path, the runner (arrow and hare) is collapsing infinite possibilities into lived/actual experienced. No geometric series, or mathematical expression can account for, or predict the inherent free-will within each electron, atom, person or planet. However, the downward causation (the constrains imposed by the collective of which each is part) does provide predictability (fate), but only to the extent that the part chooses to enjoin that collective (atom, molecule, community, planet, probability). Thus, while there is overall predictability (of the collective/probability) there is very little predictability of each part within it. Insurance companies work on the same principles. Individual-unpredictability, collective-predictability, but where insurance companies go wrong is to get too stuck on the predictability of large collectives, ignoring the rising potentials and energies that ripple through populations, and physical systems. In other words, most sciences are still in the stone-age, ignoring the rising, fluid potentials from which physical systems emerge and self-organise, with intent. Hence Darwinian theory is partially correct, but mostly misses the point, like the MWI adherents, and the vast majority of those in the scientific community. Lamarck will be seen to be well ahead of his time, but also he was very limited in his understanding, from what I understand. Neuroplasticity and other developments will reveal the underlying physioplasticity of brains, cells, bones, atoms, molecules, rocks and the physical system. It's just a matter of when, not if. Hopefully in my lifetime.
 * Steaphen (talk) 23:34, 10 November 2009 (UTC)
 * [update] a good friend with whom I share and discuss the finer implications of quantum theory suggested the above could have been worded more descriptively, by basing the description on the wave-particle model: 'particle in one sense, wave in another'. In other words, instead of 'concrete but fluid', it would be better to say 'concrete in one sense, fluid in another.' Similarly, 'possible from one perspective, actual from another perspective'. That way, the paradoxes in everyday life more fully echo that of the fundamental 'wave-particle' paradox.Steaphen (talk) 12:09, 16 November 2009 (UTC)
 * It it good to see that you had someone explain to you the concept of wave-particle duality. You might be interested to learn that a Danish scientist called Niels Bohr showed in the 1920s that the wave-particle duality is actually no paradox. While it might appear paradoxical to a 19th century scientist, in quantum physics the wave and particle descriptions are complimentary. There is no 'wave-particle paradox', but wave–particle complementarity. Ansgarf (talk) 23:52, 16 November 2009 (UTC)
 * YOu know Ansgar, if I hadn't met you I would have sworn you were a comedian. Your responses are highly entertaining. They do make me work tho' -- I continually find myself asking "he can't be serious, can he? He must be joking, surely?"
 * Yes, I'm grateful that I've had someone explain wave-particle duality to poor little me. Dear me, I'm a sad case. Ansgar, tell me, what exactly do you think is happening when the particle displays wave qualities? What exactly do you think is happening in single-photon double-slit experiments wherein a single particle travels through both slits, at the same time. Pray tell, how do particles do that? So that's not a paradox to you, that a single particle is in two places at once -- two contradictory aspects of reality? Ah, I know, yes they're just probabilities distributions. We don't need to understand what those witches, whoops, 'distributions' are really made of, do we? No siree. We'll just quote something from The Gospels (of Science). Steaphen (talk) 02:38, 17 November 2009 (UTC) (btw, has Mastercard trademarked "priceless" yet -- if not, then may I say, 'priceless'.
 * I am glad that you could see the humour in the amicable but figurative pat on the shoulder for you efforts in quantum mechanics. There is nothing magic about distributions; they are a mathematical abstraction that seems to work, even though it may be difficult to give a realist interpretation to them. If you don't like probability distributions as name, try quantum superpositions. You seemed to be happy with that before.
 * Either way, it shows imho a lack of imagination to insist on analogies from macroscopic level to describe the quantum level, like "wave" and "particle", and then even to expect that they behave exactly like their analogue on macroscopic level. The photon particle does not show wave qualities, and the photon wave does not show particle properties, but it is still the same photon. That is what experiments like the double-slit experiment show. Not much to do about it.
 * This discussion is imho somewhat off-topic, since QM has little relevance to ZP. Before you jump on the chair, I know that you think otherwise. We are discussing QM in this thread in part because you have brought it up consistently to support your position. You used it as argument that was true beyond doubt. Therefore, your accusation that I simply quote from the The Gospels (of Science) doesn't faze me much; it is just an instant of a pot calling a kettle black. And that I am actually referring to the content of the theory doesn't bother me either, to be honest. Ansgarf (talk) 22:43, 17 November 2009 (UTC)

arbitrary break 2

 * "What exactly happens when the runner (and every part thereof, including all the atoms in his/her body) begins moving and moves to say, position 0.01 x 10-50 metres?" and can be verified by experimental evidence sufficient to confirm your theories?
 * What mathematical or geometric expressions can fully predict and track the path of said runner (and any part thereof)?
 * What theory can you offer that is congruent with the experimental evidence?
 * Whatever that theory, how does it supersede that of quantum theory that has relevance and proven success in providing frameworks of description for such small increments of movement?
 * Again, how do your suppositions reflect actual reality, rather than being the "simple ideas of geometry" that Feynmann dismissed around 40 years ago.
 * Save your answers, and references to Reliable Sources for the mediators. Cheers. Steaphen (talk) 00:12, 18 November 2009 (UTC)
 * I'll still repeat my answer. At these levels the best description we have is quantum mechanics. And I doubt that you will be experimentally refute any predictions at that level unless the predictions are way off. Since this is the same answer I gave before, I assume that we can close this thread.Ansgarf (talk) 00:39, 18 November 2009 (UTC)
 * "At these levels the best description we have is quantum mechanics" Correct, therefore infinite-series are emphatically NOT able to resolve Zeno's Paradoxes, at least as far as our "best description" is concerned. Thus, "using ordinary mathematics ... we CANNOT calculate where and when a runner will overtake the hare" or words similar. Took long enough, but we got there. Please now correct the main article accordingly. Cheers, Steaphen (talk) 00:50, 18 November 2009 (UTC)
 * Our best description may not use "ordinary mathematics", but uses among others "advanced calculus" and a "continuous time partial differential equation" commonly known as "Schroedinger equation". But I mentioned that before. And It will be mentioned in mediation. Ansgarf (talk) 01:15, 18 November 2009 (UTC)
 * I ignored your comments because they are irrelevant to the issue: that of infinite-series providing real, substantial and meaningful solutions to Zeno's Paraodoxes. They can't and they don't. Your deflection to calculus being used in wave-functions or whatever else is irrelevant. Such statements (as above) do nothing to support your contention that infinite-series can or do solve Zeno's Paradoxes. Nothing. Zero. Providing a reliable source (someone who will commit career suicide) would be a good start. Then how you've successfully disproved the uncertainty Principle, and then the explanation of how the experimental evidence which does not support infinite-series solutions, is accommodated by your theories. Cheers, Steaphen (talk) 01:59, 18 November 2009 (UTC)
 * You got the implication the wrong way. Nobody tries to disprove the uncertainty principle. It actually hurts your argument, since no experimental setup can disprove any claim in the order of 10-52. And if you accept that quantum mechanics describes best what is going on at quantum level, you accept that the change of the state is best described by a continuous time partial differential equation. This is an infinite-series solution in almost every aspect. You told us before that you want to wait until mediation to provide references, it would courteous for you not to ask for references either. Because I am still awaiting for  you to provide for a reliable resource to the effect that we cannot analyse motion using ordinary mathematics. But I can wait. Ansgarf (talk) 03:25, 18 November 2009 (UTC)

The simple answer is "nobody knows" & theoretically whatever is offered as explanation can never be verified by experiment - so we use models that we already have to "model" it. Your request for "experimental proof" is inconsistent with the theoretical aspects of QM. Science does not invent novel models, especially not ones that involve infinite non-spatial space, to model what happens  --JimWae (talk) 00:20, 18 November 2009 (UTC)


 * Dear JimWae, until such times as you can provide a Reliable Source in support of your "pet theories", the statement "Using ordinary mathematics we can calculate ..." has to be removed or at least modified to clearly state the bias and assumptions upon which that statement relies. This is why mediation was called. No point in further arguing your point. Let the mediators provide the discipline that has clearly be missing thus far. Cheers, Steaphen (talk) 00:54, 18 November 2009 (UTC)


 * btw, Ansgar, here's a thought or two I think you'll appreciate: The framework I've provided above will help you cut through the nonsense of new-age beliefs, as quickly and as easily as a "hot knife through butter" -- e.g. easily dismissing many of the beliefs surrounding enlightenment, and that of 'transcending the ego'. Apply the paradox, and all begins to start making sense: religion, new-age, science, politics. What you won't like is that it also enables you to see through and beyond the highly limited, and increasingly pernicious beliefs of science. It doesn't discriminate. Cheers, Steaphen (talk) 20:36, 11 November 2009 (UTC)
 * This has very little relevance to the issue at hand, namely whether you can have an infinite series when space/time is discrete. I understand the benefit of starting from a paradox, a set of contradicting assertions. It allows you, like a turncoat, to be always right. This works well as long as you can conceal the inherent contradiction in your positions. Your essay on free-will, causality, individual vs collective, potential, ripple effects, stone-age scientists, Darwin, Lamarck, and neuroplasticity is consequently just a smokescreen to this effect. Also, in your attempt to play on the man you entered your authority as belief doctor as argument into the debate.  Which means that it becomes debatable. Frankly, I would find this yet another distraction, so I hope that we won't in the future have to debate my beliefs in religion, new-age, science, politics, nor your qualification to doctor them. Ansgarf (talk) 22:51, 11 November 2009 (UTC)
 * Crickey Ansgar, have you been eating too many coffee beans, or wot? It has no relevance to the issue at hand (other than providing an exceptionally robust framework by which to realise the infinite-series solutions for Zeno's arrow are dead and dusted. But that framework is not relevant to the issue at hand, I agree). Maybe you should you start meditating, but if you've already started, I think you should stop :) Steaphen (talk) 03:00, 12 November 2009 (UTC)
 * I may or may not have eaten too many coffee beans, I have or may not have started meditating. Do not blame me for the fact that your framework has no relevance to the issue at hand. It does not say a single word about whether you can define an infinite series on non-zero distances on a finite domain. And this is no surprise, since have you been demonstrably unable yourself to say something on the topic. All we get is smoke and mirrors. Or do you have an answer? It would really help if you had one. If you had one. Ansgarf (talk) 03:21, 12 November 2009 (UTC)
 * and this has supposedly all proven by quantum theory AND accepted by physicists? How can you seriously think that the article could reflect any of this pet theory of yours? And the space between space is ... what, non-space? --JimWae (talk) 23:38, 10 November 2009 (UTC)


 * Not much interested in what the majority crowd thinks. My "pet theory" is not at issue here. I've simply provided some context for you and Ansgar. The fundamental reason for the mediation is the inability to apply infinite-series to solve Zeno's Paradoxes. The rest is all a side-issue, and ultimately of no relevance to the mediation. But in not having shown appreciation for some insights that will last you a lifetime, I'll respectfully stay focused on the mediation issue. Steaphen (talk) 23:50, 10 November 2009 (UTC)
 * It still is a fair question to ask for experimental evidence. You are providing a new interpretation of quantum theory, while for the accepted interpretations it is still debated whether it would be possible to distinguish them experimentally at all. And it seems that you have been consistently claiming that your interpretation has been experimentally confirmed, don't you? You have been touting the need to experimentally confirm even what most would call 'thought experiments'. Ansgarf (talk) 00:22, 11 November 2009 (UTC)
 * It may indeed be a fair question, but it is irrelevant to the task at hand. Furthermore, no proof is required for that which is not in contention. What is in contention is the assertion by JimWae that "using ordinary mathematics we can calculate..." That statement is incorrect. There has been no Reliable Sources confirming that statement. As before, any such statements necessarily disqualify the Uncertainty Principle (as the infinite-series solutions absolutely require a strict, hard 1:1 physical correspondence of position with physical things (i.e. an arrow must be physically measurable at EVERY point in its progression.) This is clearly contrary to quantum mechanics, and thus untenable. Once again, the task at hand is cleaning up the main page. I've provided additional commentary for your amusement or ridicule, whatever, but the original call for mediation was requested to clean up the main page. The mediation request has nothing to do with my "pet theories" (an amusing phrase by JimWae, given that everyone operates by their own "pet theories", without exception :) Steaphen (talk) 05:43, 11 November 2009 (UTC)
 * I get the impression that you try to duck the question, by asking for references. Not sure if you accept the article on the Correspondence principle, but at least it will give you some leads. Since we are at it, do you have a reference to a physicist who says that we cannot analyse motion using infinite series? Ansgarf (talk) 07:17, 11 November 2009 (UTC)


 * Ansgar, as put to JimWae, let the mediators provide some discipline, and closure. Cheers, Steaphen (talk) 07:28, 11 November 2009 (UTC)
 * It appears you find it opportune to ask us for references while mediation is pending, but when the favor is returned you decline to give a reference while mediation is pending. Ansgarf (talk) 07:38, 11 November 2009 (UTC)
 * "At these levels the best description we have is quantum mechanics" Correct, therefore infinite-series are emphatically NOT able to resolve Zeno's Paradoxes, at least as far as our "best description" is concerned. Thus, "using ordinary mathematics ... we CANNOT calculate where and when a runner will overtake the hare" or words similar. Took long enough, but we got there. Please now correct the main article accordingly. Cheers, Steaphen (talk) 00:49, 18 November 2009 (UTC)
 * What quantum mechanics shows is the incredible power of infinite series solutions. Repeating your claim to opposite is repetitive. Why not wait for mediation. You have posted the same claim about three time today, and I have given you the same answer as often. While this all is seriously off-topic.Ansgarf (talk) 03:31, 18 November 2009 (UTC)
 * Dear Ansgar, I'm curious to what extent I'll be entertained by your (must-have-the-last-post) reply.
 * (In Newtonian speak) Mathematical calculation = actual physical location and speed (no ifs, buts, or maybes). Absolute, perfect determinism. That is what infinite-series requires. Absolute, perfect 1:1 correspondence of calculated location/speed with actual physical location and speed. No exceptions. Infinite-series (and the absolute determinism upon which it is based) disallows Heisenberg's "uncertainty" principle in regards to location and speed (momentum). Either Heisenberg is wrong, or infinite-series cannot be used to resolve Zeno's Paradoxes. As for infinite-series of state-vectors or whatever else, totally irrelevant. We are talking reality here, and theories that are congruent with that reality, not idle conjectures as to what might be happening, but cannot be substantiated in fact. If your theories cannot be shown to be congruent with actual reality, and yet you argue for them, you are no better than the witch-hangers, or heretic-burners of yore. The main page is wrong. There are simply no grounds by which anyone (even in the proposed section) can argue that "using ordinary mathematics we can calculate ..." As far as 'off-topic' reference is concerned, the topic here is Zeno's Paradoxes, and the errors and bias on the main page thereof. If you believe your comments are off-topic then you are welcome to discontinue making them. Cheers, Steaphen (talk) 07:11, 20 November 2009 (UTC)

Hi Steapehn, When I said that we should wait for mediation, I was actually echoing your earlier statements made on at least 10, 11 and 18 November 2009 that we should wait for the mediators. It is obvious that we will probably just repeat our arguments until then.

That said, I am happy to repeat my arguments again.


 * First, you are barking up the wrong tree. We are looking at the following case distinction
 * (P1) Assumed a dense model of time and space then (C1) the infinite series converges
 * (P2) Assumed a discrete model of time and space then (C2) there is no infinite series.
 * Your argument, Steaphen, is that you cannot use conclusion C1 given premise P2. That is correct, but nobody does this. (BTW Jim, I accept that C1 doesn't address all aspects of the paradox.)

We have been over most of these arguments before. I'd be happy to wait for mediation to start, I'll then repeat all of these arguments. But if you cannot wait, I am happy to repeat them again here. I actually enjoy to point out some facts about QM that you apparently didn't know yet, and I did learn something in the process as well. Ansgarf (talk) 04:48, 21 November 2009 (UTC)
 * Then you seem to misunderstand the uncertainty principle. It is about the measurement of position and momentum at the same time, but it says nothing about the position itself.
 * Heisenberg never said that you cannot use infinite series to describe position or motion. That is actually an outrageous claim, since Heisenberg's matrix mechanics uses infinite dimensional matrices, and all operations on them are by definition infinite series. Furthermore, the matrices that he used to define position and momentum evolve in continuous time. And finally, these matrices satisfy the classical equations of motion. Of course, the equations also entail that you cannot measure the position with infinite precision, but that is an orthogonal issue.
 * We agree, you cannot apply the naive notion of an "actual position" to the quantum level. The best you have are distributions. The best abstraction for the "actual position" is the expected value of that distribution. Which is a point in space. Since the Schroedinger's equation is continuous time, the trajectory of the expected value ("actual position") will be continuous time as well. If you accept QM, you implicitly accept a continuous time trajectory of the expected value ("actual position"). If you like it or not.
 * I pointed a fair number of times to the different interpretations of QM. You should have noticed that that is is still disputed what the "reality" of the equations is. The only thing that has been established is that the mathematical model, which QM is, works well and predicts experimental observations. And that model uses state vectors and infinite-series. I know, you choose to ignore this, but it doesn't make it less true.
 * Dear Ansgar, whether space-time (and any inhabitant thereof) is dense or not, is irrelevant to the mediation case. As are assumptions P1 or P2. As is whichever interpretation of quantum mechanics you care to name. It's all irrelevant to the issue of whether we can "using ordinary mathematics to calculate ...". What is relevant to the issue of Zeno's Paradoxes is this: whether you can use a calculation (by any means) to fully and totally determine the speed and location of physical things. Infinite-series solutions require that you can. Quantum mechanics says you can't. One says yes (using 'simple ideas of geometry extended down into infinitely small space' to calculate when the runner overtakes the hare). The other (via experimental evidence and quantum theory) says no, you can't. Not now, not ever.
 * Your demonstrated refusal to apply scientific-method of matching theory with observable reality speaks volumes about the extent to which you (and the general scientific community) have lost touch with reality. So serious and deep is that disconnect I genuinely believe our race is imperilled as a result. Because of that disconnect we are building momentums into the physical ecosystem that will be difficult to turn around. Having said that, when facing great adversity the human spirit can achieve the extraordinary. So while exceptional challenges and changes are headed our way, we can prevail. Cheers, Steaphen (talk) 12:34, 21 November 2009 (UTC)
 * An accepted idea that explicitly states that you can use ordinary mathematics is relevant to the issue whether you can use ordinary mathematics. Choosing to ignore this is simply wilful ignorance, which doesn't strike me as very scientific either. It seems like you retreated to the claim that your interpretation of the uncertainty principle proves that the mathematics of QM is impossible. Which ironically means you argue that because of P2, you cannot C1.
 * The uncertainty principle affects all measurements. But it does not limit mathematics. If you add up one and one of the natural numbers you'll get two. Because these numbers are abstractions the uncertainty principle does not apply. And it applies neither to infinite-series calculations in the real or complex numbers. If you then apply the result to distances in reality and measure in meters, you will get a matching result within the limits of uncertainty. QM provides a calculation to fully and totally determine the speed and location of physical things and it also provides a model of what to expect once you perform an experimental measurement. It is a mathematical model that has been experimentally confirmed. The math does not require you to measure each and every step. Actually the mathematical model of QM will tell you that the result depends on how often and when you measure. Which has been experimentally confirmed too. However, this and should be and is covered under the topic of uncertainty.
 * Have you any evidence for any of these claims in the latter paragraph? That I imperil the future of humanity? Or even what my position on large infrastructure projects is? Or even that I do not apply the scientific method? But before you provide evidence, could you elaborate why your remarks weren't off-topic.Ansgarf (talk) 00:48, 22 November 2009 (UTC)

Random break
You are making 2 major mistakes. You are assuming that if space were continuous (or even dense) that calculus would defeat Zeno's paradoxes. It does not, but at least you would not be alone in that misunderstanding of Zeno. Your second mistake is so illogical that it is hard to make any sense at all of your position. You are saying that discrete space still leaves an infinite series of distances between two points in space - that this inter-spatial, non-spatial, infinite "space" is somehow filled with something. The existence of a paradox does not justify bewildering pet conjectures, nor can such unsourced conjectures be represented in the article --JimWae (talk) 00:00, 11 November 2009 (UTC)


 * Whatever. Let the mediators provide some discipline, and closure. Cheers, Steaphen (talk) 00:07, 11 November 2009 (UTC)

Formal mediation requested.
Dear Ansgar, and JimWae

You have both been included as relevant parties in a request for formal mediation regarding the issue of infinite-series in Zeno's Paradoxes.

Details: Requests_for_mediation/Zeno's_paradoxes

Please respond as appropriate.

Looking forward to developments.

Kind regards, Steaphen (talk) 02:26, 8 November 2009 (UTC)

Moving over to the Formal Mediation pages.

1. Steaphen's Resources and preparations for Formal Mediation Case Steaphen (talk) 22:03, 26 November 2009 (UTC)

Please confine discussions to editorial concerns
Most of the above discussions involving user:Steaphen, however interesting, are essentially off-topic and a misuse of this talk page. See Talk page guidelines, the first sentence of which reads:
 * The purpose of a Wikipedia talk page is to provide space for editors to discuss changes to its associated article or project page. Article talk pages should not be used by editors as platforms for their personal views on a subject.

Can we please keep the discussion focused directly on specific proposed changes? Paul August &#9742; 23:23, 17 November 2009 (UTC)


 * "Most of the above discussions involving user:Steaphen, however interesting, are essentially off-topic." Interesting. Question: would the responses by others to my posts be entirely as per the above guidelines? Statements such as "Using ordinary mathematics, we can calculate ..." (one of the main issues raised in the mediation) imply or carry underlying assumptions. Are you suggesting these assumptions are correct, and therefore need no clarification, or that there is no need of Reliable Sources confirming their validity? The fundamental unsupportable bias on the main page remains uncorrected. If you have Reliable Sources that can support or address what I perceive as errors, please provide them -- sufficient to resolve the presumably "off-topic" questions I have raised. Cheers, Steaphen (talk) 00:25, 18 November 2009 (UTC)
 * I admit that most of my responses to Steaphen are only on-topic with respect to his objections, but off-topic with respect to ZP. Ansgarf (talk) 00:42, 18 November 2009 (UTC)

1>Can we at least drop all the personal chat & comments about faith? 2>Where is your proposal for a new wording? --JimWae (talk) 00:29, 18 November 2009 (UTC)


 * I don't believe it is necessary for me to provide "new wording". I'm simply pointing out that statements (and the underlying bias) on the main page is unsupported by Reliable Sources, and needs to be either entirely removed, or changed appropriately to reflect a lack of bias/POV (which is based on the assumption of physical continuity). Cheers, Steaphen (talk) 00:37, 18 November 2009 (UTC)

To clarify, my remarks apply equally to all editors. Paul August &#9742; 00:54, 18 November 2009 (UTC)


 * Thank you Paul. Albeit for some digressions, my main focus has and remains the errors on the main page. Formal Mediation has been called (and if necessary, arbitration as well) to address those errors. Cheers, Steaphen (talk) 00:58, 18 November 2009 (UTC)

Resources and preparations for Formal Mediation Case
Perhaps it will expedite the Mediation case if relevant Reliable Sources are included here. Please do not add comments or opinions in this section. This section is for Reliable Sources relevant to this mediation.

To begin:

Physical movement (of Zeno's arrow, Achilles, tortoise etc), based on infinite-series solutions, is believed (and required) to be contiguous and continuous.


 * Reliable Sources confirming that space-time and/or physical movement is not continuous and contiguous are:

"according to the quantum theory, movement is not fundamentally continuous." David Bohm, Wholeness and the Implicate Order, Routledge, London 1995, page 202. (italics/emphasis by Bohm) "that space is continuous is, I believe, wrong." Professor Richard Feynman, The Messenger Series: Seeking New Laws (according to some estimates) "our universe is flickering on and off every 5.3 x 10-44 seconds", Norman Friedman, The Hidden Domain: Home of the Quantum Wave Function, The Woodbridge Group, Eugene OR 1997, page 165. "At the smallest level of space-time-matter, space-time is continually fluctuating—creating momentary bubbles of matter, which just as quickly vanish into nothingness again." Fred Alan Wolf, Parallel Universes, Paladin, London 1991, page 188 "Zeno’s Dichotomy and Achilles-and-Tortoise Paradoxes rule out the possibility of continuous space-time." Chen, Gikuang Jeff, Resolving Zeno's Paradoxes with Discrete Space-Time (12/04/2005). Available at SSRN: http://ssrn.com/abstract=1133624 "to see the arrow move as a series of continuous dissolving movie frames, we must view many more than the modern filmaker's usual twenty-four frames per second. We need an infinite number of frames passing before our eyes each second. So dividing up motion into infinity is really no different than adding up to infinity. This subtlety eluded Aristogle and everyone who came after him for the next two thousand years and more. By assuming that the arrow's motion was continuous, it was natural to imagine continuity as 'made up' of an infinite number of still frames, eve though we would never attempt to make such a movie picture. We just believed that 'in principle' it was possible. By 1926 that hope was demolished. Werner Heisenberg, the young physicist, who demolished it, was later to be awarded the Nobel prize in physics for his realization that Zeno was correct after all. Heisenberg's Principle of Indeterminism (or Principle of Uncertainty, as it is often called) reaffirmed Zeno's objections that "an object cannot occupy a given place and be moving at the same time." Heisenberg recognized that observation, as we actually experience it does not allow us to analyze motion on to infinity. Sooner or later we see that our activity introduces discontinuities in whatever we are observing. These discontinuities are fundamental to the new physics of the twentieth century." Dr Fred Alan Wolf, Taking the Quantum Leap, Harper and Row, New York, 1989, p.21. (winner of the American National Book Award, 1982).  "no metaphysical sense can be made out of mathematical sense and any claim to the contrary is unjustified. And further that any resolution to Zeno’s paradoxes, if it is to “hit the point”, must indeed make metaphysical sense." Alba Papa-Grimaldi, Why Mathematical Solutions Of Zeno's Paradoxes Miss The Point: Zeno's One And Many Relation And Parmenides' Prohibition The Review of Metaphysics 50 (December 1996): 299-314.   [tba] </li> <li> [tba] </li> </ol>
 * Call for Reliable Sources confirming that space-time and/or physical movement is continuous and contiguous (upon which infinite-series solutions are reliant)


 * Dr Fred Alan Wolf (winner of the 1982 American National Book Award for Science) states that "observation, as we actually experience it does not allow us to analyze motion on to infinity" -- what Reliable Sources confirm that we can, contrary to Dr Wolf's statement, "analyze motion on to infinity" as is done with infinite-series solutions?


 * Please list Reliable Sources below:

<ol> <li> [tba] </li> <li> [tba] </li> </ol> Steaphen (talk) 23:29, 24 November 2009 (UTC)

Comments and opinions on the above section

 * This an interesting development. [| Last week] you were denying any relevance for the mediation whether space/time is continuous and right now you seem to collect references to interpretations that either assume (P1) space/time is continuous, or (P2) it is not. A small semantic issue by the way, all that reliable sources do is to argue what would be the most suitable model. That is still one step away from confirming the reality of continuous/discontinuous space time. But I am happy to provide references for either side. Your effort just confirms that you do want to argue that since space/time is not dense, the calculus solution is wrong. Ansgarf (talk) 13:01, 24 November 2009 (UTC)

Discrete-time model
 * G. Date. A Discrete Time Presentation of Quantum Dynamics. 	Class.Quant.Grav.20:303-316,2003
 * Bender et al. Discrete-time quantum mechanics. Phys. Rev. D 32, 1476 - 1485 (1985)
 * Y. Jack Ng et al. Probing Planck-Scale Physics with Extragalactic Sources? ApJ 591:L87-L89, 2003

Continuous-time model Hope it helps. Ansgarf (talk) 13:37, 24 November 2009 (UTC)
 * Bohm D. A suggested interpretation of the quantum theory in terms of “hidden variables”, Phys. Rev. 85, 166(I) – 180(II), 1952.
 * Roberto Ragazzoni, The Lack of Observational Evidence for the Quantum Structure of Spacetime at Planck Scales. ApJ 587 L1-L4 (2003)


 * Just a few more remarks.
 * This list uses the term "discontinuous" rather loosely. A set can be piece-wise continuous, i.e discontinuous, even in continuous time. This might be what the Bohm quote means. This is something different from the notion of a "flickering movie" which means that the time is discrete, disconnected, non-convex, but closed. This is something different, from the notion of "foam", which can mean a set that is connected, closed, but non-convex. You can define continuous functions on "foam".
 * Some of the references, like Papa-Grimaldi, just highlight that mathematics do address some of philosophical problems connected to Zeno's paradox. This is not contended. The article reflects this (3rd paragraph).
 * None of the references actually states that you cannot use "continuous" mathematics to describe motions. Bohm's mechanics for example is continuous time, and Date states in the abstract of his paper, that his discrete time model is consistent with the continuous-time description (in a more elegant way than other models). And others on the list made similar statements; that a continuous time model is appropriate.
 * None of the references state that since time and space might be discontinuous in QM means that solutions to infinitive-series in dense time are mathematically incorrect or impossible.
 * Most of the references essentially just undermine Zeno's assumption that between any two points there exists another. But this case is already addressed in the article (last paragraph).
 * This list of references suggest that the contention is whether mathematics solves every aspects to the paradox, or whether time and space is "continuous". This is incorrect. The contention is whether you can use classical mathematics to compute when the Achilles passes the tortoise. These references say not much on this issue if anything at all. about this issue. Steaphen is just jumping to conclusions. Ansgarf (talk) 01:22, 25 November 2009 (UTC)

I'd rather have resources for the article
At least something positive to come out of this endless nonsense: Finally we start to see citations. It's a start. Regrettably, this still misses the point of WP:RS: The sources must be relevant to Zeno's paradoxes, not to discussing the subject. As of this writing, this applies to all citations in the above section (and the one split from it), except the Chen reference, which was published on SSRN, which does nothing to support the WP:RS claim. Also, Chen has no scientific credentials I could find. Friedman is a vanity-published quack. None of the other citations give any clue that the arguments/positions were applied to Zeno's paradoxes.

Which leads us to a bit of windfall: Checking up on Chen, I stumbled across this paper, which is also self-archived at Papa-Grimaldi's homepage. That it is catalogued on PhilPapers doesn't mean too much, so is Chen's paper. The author was possibly a student of Tim Crane, other than that I couldn't find much on him/her. But I think the fact that it was published in Review of Metaphysics is sufficient to establish it as reliable primary source. As long as WP:DUE is taken into consideration, a brief mention would be okay with me.

Regards, Paradoctor (talk) 15:23, 24 November 2009 (UTC)


 * re "Friedman is a vanity-published quack" -- I've had to alert the Wikipedia board of your comments here, suggesting you've opened Wikipedia to possible Libel action. Friedman holds a degree in physics and a masters degree in engineering. I would think the above constitutes defamation, and good grounds for him to sue Wikipedia.


 * As for your other comments, they essentially disqualify you from further serious consideration as a competent editor. Cheerio Steaphen (talk) 20:11, 24 November 2009 (UTC)


 * Sure, go ahead. :) Paradoctor (talk) 21:52, 24 November 2009 (UTC)

Aaaah, citation #6: That's the kind of contribution we can talk about, Steaphen. Source with quote establishing relevance, and published by a professor of physics, no less. Regrettably it's not a reliable source in my understanding. According to Wolf's own statement, his expertise is "high atmospheric particle behavior following a nuclear explosion". He lists nothing else, Scirus found nothing, Google Scholar found his Ph.D thesis and three papers on quantum consciousness. Taking the Quantum Leap: The New Physics for Nonscientists is not a scientific publication, I'm afraid. But you're definitely heading in the right direction. Regards, Paradoctor (talk) 23:23, 24 November 2009 (UTC)
 * Citation #6 is looking at the discontinuities introduced by observation. This is not really contended. This paragraph doesn't say anything about whether continuous models are suitable to model motion. For example, a few pages later in the same Chapter 2 Fred Alan Wolf says the following:
 * "'I would turn out that both 'truths' of Zeno and Aristotle were correct. Motion was continuous and smooth, provided it was unseen. Motion was discontinuous, whenever it was observed, provided we looked hard enough to see it.' (Dr Fred Alan Wolf, Taking the Quantum Leap, Harper and Row, New York, 1989)"
 * It seems that even Fred Alan Wolf accepts that the motion is correctly modelled by the continuous-time Schroedinger equation, while measurement are best modelled as discontinuities (collapse of the wave function).
 * BTW: The last  paragraph does deal with the case that time is not "continuous". Ansgarf (talk) 23:45, 24 November 2009 (UTC)


 * Once again (ad infinitum?), your inability to perform (what I find) elementary analysis is the problem here. The fact that continuous-time functions/calculus/infinite series can be effectively and meaningfully applied to wave-function POSSIBILITIES has never been questioned, denied or implied. What has been consistently and repeatedly asserted is such functions have no meaningful relevance to actual PHYSICAL movement, beyond crude, Newtonian approximations (that are deficient by infinite orders of magnitude in dealing with the paradoxes). The mathematics of QM or that of infinite-series, or calculus, or ANY form of mathematics is not in contention here, only its application to resolving the very real, everyday physical phenomenon of physical movement (of Zeno's runners, arrows, hares etc.).
 * In summary, there is NO known mathematics that can account for physical movement (in absolute detail). None. If there were, it could be applied to the 'collapse of the wave-function' to precisely predict the "collapse" of possibility into lived, experienced physical reality. The statement "Using ordinary mathematics we can ..." is simply wrong. We can approximate, but not calculate. That is the fact of the matter. The main page contains bias and errors. Hence the necessity for Formal Mediation (and in all likelihood, arbitration) Steaphen (talk) 00:11, 25 November 2009 (UTC)
 * I was not just simply repeating myself, I was given an example of an authority, Fred Alan Wolf, claiming that motion can be smooth and continuous while it can also be discontinuous. Interesting that you overlook this quote and simply repeated you old argument.
 * To the point, I'm happy to repeat my answer as well. Nobody is actually questioning the uncertainty principle. And I agree, you could mention uncertainty each and every time somebody makes a claim about a physical quantity. You won't be able to claim with absolute certainty that the Eiffel Tower is 324m high. Granted. But (1) there is nothing special about Zeno's paradox to mention it here; mention it in the article on the Uncertainty Principle.  (2) The uncertainty principle does not apply to mathematical or philosophical arguments, and Zeno's argument is foremost a mathematical and philosophical argument. Ansgarf (talk) 01:43, 25 November 2009 (UTC)
 * "Zeno's argument is foremost a mathematical and philosophical argument" -- says WHO?Steaphen (talk) 01:46, 25 November 2009 (UTC)
 * "Motion was continuous and smooth, provided it was unseen". Correct. Motion (in or through invisible, unseen Hilbert space/multiverse, possibilities) is infinitely smooth, but not in actual, observed reality. Your continued inability to apply scientific-method of matching theory with verifiable, observable reality borders on the stupendously stupid. And to think I've been described as someone who does not easily suffer fools. Let all be witness that I must be making progress. :) Steaphen (talk) 01:56, 25 November 2009 (UTC)
 * I'd say it is general consensus that Zeno was a classic philosopher who used an argument from an infinite series to show that the concept of motion is impossible.
 * I disagree with your interpretation of the Wolf quote. Wolf didn't distinguish between "observed reality" and "possibility", he was distinguishing between motion before and after measurement. Yes. We can disagree on this for the time being. Let others decide. I also let other decide who is stupid here. But I welcome that the mathematics is not longer in contention, but forgot to add that QM is more detailed than Newtonian mechanics, but that doesn't mean that there is an infinite order of inaccuracy. And both can be used to very real, everyday physical phenomenon of physical movement. But that aside, you keep asking for the meaning in actual physical terms  of the models in the  sub-quantum domain. You just have to accept that there exists no single accepted realist interpretation of QM. And that many physicists actually will only give you an instrumentalist interpretation. I know, you really want a realist interpretation. Sorry. Ansgarf (talk) 02:07, 25 November 2009 (UTC)
 * You know Ansgar, you're not a bad bloke, as we say in Australia. I'm presently under pressure to finish marking some uni assignments which leaves me a little terse, so I'm not particularly gracious or tolerant at present.. (this dialogue represents a distraction from the work at hand ...)
 * "'mathematical “solutions” miss, and always will miss, the point of Zeno’s arguments. I do not think that any mathematical solution can provide the much sought after answers to any of the paradoxes of Zeno. In fact all mathematical attempts to resolve these paradoxes share a common feature, a feature that makes them consistently miss the fundamental point which is Zeno’s concern for the one-many relation, or it would be better to say, lack of relation.'"
 * Alba Papa-Grimaldi, The Review of Metaphysics 50 (December 1996): 299-314.(see above for references).Steaphen (talk) 02:18, 25 November 2009 (UTC)
 * Interesting quote. You could have quoted this Wikipedia article as well. It says Some philosophers claim 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.. Papa-Grimaldi does not say that you cannot use mathematics to address physical movement, he says that mathematics does not address philosophical problems. Ansgarf (talk) 02:48, 25 November 2009 (UTC)

Steaphen, I'll heed the advice to remember the compliments, and forget the insults. Thanks. Luckily, I am done marking university assignments, and I am off for some time. Since this discussion will probably continue, just a few points I like to make with respect to possible references. It would be nice to see some references on the latter, rather than on the former three. See you in a week. Ansgarf (talk) 03:52, 25 November 2009 (UTC)
 * It is actually not contended that some interpretations of QM do not use continuous time or space.
 * It is actually not contended that some, maybe important, philosophical problems are not addressed by the calculus solution.
 * It is also not contended that the uncertainty principle means that certain physical quantities cannot be measured up to an arbitrary precision.
 * It is contended that the use of ordinary mathematics, with or without infinite series, is appropriate to describe motion of the objects mentioned in Zeno's paradox, in the context Zeno's paradox.


 * Dear Ansgar, unfortunately for you I still have some assignments to mark.


 * As previously explained, the de Broglie wavelength of an object (e.g. Zeno's arrow) is :$$\lambda = {h}/{p}$$ (where p = momentum). The infinitesimal precision of the object's position or any part thereof (as required by infinite-series 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).


 * BUT as $$\lambda$$ approaches zero (to ensure a short sharp pulse with infinite precision, exactly matching that of infinite-series), p (momentum = mass x velocity) approaches infinity. Thus, to precisely predict (calculate) an object's location at some arbitrary point in time t (as is expected using Newtonian mechanics/infinite-series), requires the object to have infinite mass, and/or infinite velocity. Despite the fact that the calculated wavelength for a runner, tortoise or arrow might be immeasurably small (short), it is still some finite wave-length. This finite-wavelength categorically conflicts with the necessarily infinitely-short wavelength required by infinite-series. This undeniably rules out infinite-precision/infinite-series predictions of when the runner will overtake the hare. The inability to calculate (even in theory) when the runner will (precisely) overtake the hare is not due to clumsy scientists, with fat fingers using blunt measuring instruments. It's simply the result of quantum theory. And that theory matches experimental evidence.


 * Infinite-series cannot be meaningfully applied to Zeno's Paradoxes. The widely-accepted infinite-series solutions to Zeno's Paradoxes are clearly incorrect, lacking compatibility and congruency with the experimental facts and quantum theory, and the main article needs to reflect this.


 * From the Wikipedia entry wiki/Matter_wave "given the enormous momentum of a person compared with the very tiny Planck constant, the wavelength of a person would be so small (on the order of 10−35 meter or smaller) as to be undetectable by any current measurement tools" -- nevertheless, it is a finite wavelength. Infinite-series do not and cannot provide meaningful congruent solutions to Zeno's Paradoxes while ever the wavelength remains finite.
 * btw, almost forgot, for those seeking to claim the above as 'original research', it isn't. I've simply highlighted what has been officially accepted since 1929 when de Broglie was awarded the Nobel Prize in physics for the discovery that physical matter has an associated wave nature (experimentally confirmed for macro-scaled objects). Hardly original. Steaphen (talk) 13:05, 25 November 2009 (UTC)


 * Before I am really off; I did explicitly mention that the fact that you cannot measure physical quantities up to an arbitrary precision is actually not contended. I ignore for the time being that on the scale that you want to do the math, the naive notion of a position does not apply. A tortoise is not a point mass. A reasonable substitute would be the geometric center or the center of mass. Both are averages.
 * That aside, to do the math that $$\lim_{n\rightarrow\infty} \sum_{k=0}^n 1/2^{k}$$ equals 2, no measurement is necessary. You need induction. And the prediction from the infinite series that the runner will be at 2, is subject to the same uncertainty as a simple claim that the runner is at 2. At that point you might want to measure to confirm the prediction, and at that point uncertainty comes into play, including any arguments from De Broglie's wavelength. As long as it is unobserved, motion is smooth and continuous. Says Fred Alan Wolf. There is nothing special about the position of  tortoises and runners that uncertainty should to be mentioned here, but not in the entry on the Eiffel Tower. It should be mentioned, if at all in the article on the Uncertainty Principle.
 * I repeat myself, but it seems like I need to stress the point again: The inherent imprecision of measurement is not in contention, but the whether the use of ordinary mathematics is appropriate to describe motion in the context of Zeno's description of the paradox.  Ansgarf (talk) 12:45, 25 November 2009 (UTC)


 * Thank you Ansgar for confirming my posts ... "As long as it is unobserved, motion is smooth and continuous" -- as you have now confirmed, so long at the arrow, runner etc. remains invisible, you can say they are moving smoothly. Absolutely, in fact you can say anything you like about the runner, or arrow, or tortoise that remains invisible and unobserved.
 * Thank you and Cheerio, Ansgar. Blessings on your journey. Steaphen (talk) 13:31, 25 November 2009 (UTC)
 * Earlier today you told a person that they were unqualified to be an editor. This comment of yours casts seriously doubt on your qualification. You do know what type of observation is associated with the collapse of the wave function, don't you? Wolf said "Motion was discontinuous, whenever it was observed, provided we looked hard enough to see it." He didn't include the latter part of the statement for no reason. First you treated Wolf's work as if he is infallible, now you treat his work as if he is an ignoramus. He didn't mean that only invisible things move smoothly. You can probably guess how much of a tortoise you actually measure, when you see it from a distance. Thanks for the blessings.   Ansgarf (talk) 14:05, 25 November 2009 (UTC)
 * Dear Ansgar,
 * As a writer, and a consultant who's main focus is Health and Wellbeing, it behoves me to leave you with some wisdom from Michael Leunig, the Australian cartoonist. Most astute people reading this page will easily recognise that the infinite-series solutions are not substantiated by the evidence. I suppose in technical terms, one could justifiably say the theory of infinite-series is quackery, given the lack of evidence for the theory. :)
 * So where do we go from here? Obviously, as your logic and your slide-rules have failed you, there remains only one dimension to life that you have yet to engage successfully (in order to match theory with evidence). If you include "intuitive" in Michael's quote (as in "vulnerable intuitive side"), you and the vast bulk of logically-bound scientists will find an enriched, expanded, congruent felt-understanding of life. A felt-understanding that will be necessary to intuit the solutions for a world increasingly out of balance with nature. As before, blessings on your journey. Steaphen (talk) 20:38, 25 November 2009 (UTC)
 * "'… until a man discovers his emotional life, and his gentle, vulnerable side, until he gives it expression, he never will find his woman or his soul, and until he does find his soul he will be tortured and depressed and miserable underneath a fair bit of bullshit.'"

Why do I even try? Paradoctor (talk) 23:24, 25 November 2009 (UTC)

Steaphen's Resources and preparations for Formal MediationSteaphen (talk) 22:30, 27 November 2009 (UTC)