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Philosophy: Logic / Ancient B‑class Low‑importance

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Mathematics Unassessed High‑priority

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Zeno's paradoxes was a good article, but it was removed from the list as it no longer met the good article criteria at the time. There are suggestions below for improving the article. If you can improve it, please do; it may then be renominated. Review: September 14, 2006.

Archives

/Archive 1

Starting clean

I think it's worthy of a B+ Now.....64.6.88.31 (talk) 17:22, 24 May 2008 (UTC)

Let's start the talk page clean, alright?Le Blue Dude (talk) 19:05, 4 May 2008 (UTC)

It seems to be starting on it's way to being a good, simple, article again. Whoot! 21:04, 13 May 2008 (UTC) —Preceding unsigned comment added by 64.6.88.31 (talk)

I agree. The biggest problem with the old version was that it argued fervently against the calculus solution (which led me to make all those long posts on the old discussion page). Now the article treats the "issues with the calculus solution" in a fully encyclopedic manner: it states that there are people who think that there are issues, and something about why they believe that, but it doesn't take a stance in the matter. Well done. Sthinks (talk) 23:49, 13 May 2008 (UTC)

Question

"Another proposed solution is to question the assumption inherent in Zeno's paradox, which is that between any two different points in space (or time), there is always another point. If this assumption is challenged, the infinite sequence of events is avoided, and the paradox resolved."

Can someone explain how matter has the special property of being able to translate itself to an adjacent point in space from rest? I think there's an assumption inherent somewhere here. —Preceding unsigned comment added by 99.225.160.154 (talk) 05:03, 6 June 2008 (UTC)

I'm with you, and I think your question closely resembles Zeno's question as posed in the Arrow paradox. Indeed, Zeno raised some deep metaphysical questions in general about motion that don't seem to have been resolved. That is, as engineers we can talk about motion as simply being at different points in space at different points in time and, as such, we can make all kinds of predictions about when and where some object is going to be, often using calculus. However, what goes on metaphysically that makes this work is not clear at all. For one, there is the problem of the infinite sequence of points, and for another, is motion really a point-to-point kind of process? Indeed, is there even anything in reality corresponding to our notion of a 'point' in the first place? —Preceding unsigned comment added by 72.226.66.230 (talk) 13:12, 6 June 2008 (UTC)

Phraseology

"Aristotle remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances."

I would hazard to suggest rephrasing the second sentence to read "Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of divisions of a finite distance." —Preceding unsigned comment added by 128.244.96.176 (talk) 04:24, 14 June 2008 (UTC)

Painting Pictures

Mathematical solutions aside, let's see if we can't expose any flaws intuitively.

Dichotomy:

Take a video of a person walking 60 yards in 60 seconds. Start replaying the video and pause it when the person has walked half the distance. Wait five seconds. Resume the video and pause it again when the person has walked half the remaining distance. Wait five seconds. Continue ad infinitum. Will you ever see the person reach the 60 yard mark? No. Will you ever finish watching the video? No. Does that mean that the person never really walked 60 yards in 60 seconds? No.

(Note: the above assumes that you have an infinitely fast thumb on an infinitely responsive pause button!)

Arrow:

Q: What would it be like to live in a universe where there are only three spatial dimensions and no time dimension? A: Motion would be nonsensical (how can there be distance divided by time when there's no such thing as time?). This is effectively what Zeno is asking us to consider when he begs the question of time being made up of an infinite sequence of timeless "now"s (an infinite sequence of timeless "now"s can certainly be located in time - but since their temporal sum is zero is it possible for them to constitute time?). Not only that, he's asking us to accept that his thought-experiment appearance of "rest" in a timeless "now" is more real than the empirical-experiment appearance of motion in a continuous-time sensing of the world.

While we're at it, let's try another tack. The arrow paradox presupposes continuous space and an infinitely discrete time where each "now" has no temporal extent and the arrow occupies a continuous spatial extent equal in size to its physical being. Let's reverse that and presuppose continuous time and an infinitely discrete space where each "here" has no spatial extent. For any given "here" through which the arrow's (apparent) motion takes it, there is a temporal extent associated with the arrow - that is, the arrow occupies the spaceless "here" for the amount of time it takes for the (apparent) motion of the arrow to carry it through the spaceless "here". Rationalize (literally) these two artificial perspectives - the spatial extent of the arrow in a timeless "now" with the temporal extent of the arrow in a spaceless "here" - and we get a measurement of: motion. —Preceding unsigned comment added by 128.244.96.176 (talk) 05:17, 14 June 2008 (UTC)

While I've been fascinated by the idea that that the Galilean priests who refused to look through Galileo's telescope have reincarnated into contemporary times (replete with their refusal to look through the 'quantum telescope' and offer theories to match observable reality), it seems you've made an effort to look, and to question.

However, your "rationalising" is, in my opinion, the point at which you paused in your questioning. Rational thinking and reason is derived from the Latin ratio. In a sense, to reason is to separate and compare (to objectify), and such objectification (by definition) won't get you within a hoot or a holler of "timelessness" or "spacelessness" (for how are you going to separate yourself from no-thing?). But you have at least questioned without referring back to the Gospels (of "science").

While I may make light of the general refusal to "look through the 'quantum' telescope" it is, I think, highly pertinent to remember that the human race has grown through the art of looking and questioning. So for all our sakes, I encourage you to continue questioning.

Steaphen (talk) 15:12, 14 June 2008 (UTC)

"... derived from the Latin ratio ..." - and it is precisely this limited literal sense that I intended ... the ratio of the spatial extent of the arrow in a timeless "now" to the temporal extent of the arrow in a spaceless "here" is distance/time - that is, the speed of the arrow. It seems to me that Zeno strips away an entire dimension under the guise of colloquial language ("now"). I thought it only fair to complete what Zeno started by stripping away the other three dimensions and seeing where that led (whether I did a good job is open to question). And what's interesting is that in re-integrating these two orthogonal projections of space-time through an operation as simple as the ratio, we regain what Zeno asserts is lost: motion! ;-) —Preceding unsigned comment added by 128.244.96.176 (talk) 20:19, 14 June 2008 (UTC)

By my rationlization :) working with spacelessness and timelessnesss, on a logical basis reveals the following: Starting with velocity=distance/time (v=d/t), and taking d=0 (since spacelessness presumably means zero distance) and t=0 (same again for timelessness) then you are suggesting that velocity v=d/t (e.g. of an arrow) is somehow derived from dividing zero by zero?

Once again, I think most contributors on this and the preceding "archived" page have what I call "GPS" .. Galilean Priest Syndrome. Here's a thought experiment. Imagine that the priests in Galileo's time were intelligent people (perhaps equal or greater than our intelligence). Why might they have, despite their intelligence, avoided looking through the telescope? The thought experiment is basically asking why might quite sane, intelligent folk avoid the scientific method: observing evidence and then working towards a new theory to fit said evidence?

As I previously made quite clear on the "archived" page, Zeno's arrow, hare or runner MUST inevitably pass through quantum scale increments, since by definition, geometric series require continuity ad infinitum. We know from quantum physics research that simple geometric continuity ceases. Accordingly, the standard mathematical treatment as provided by others at this forum, does not fit observed reality. It is indicative of GPS that no one has once made any attempt to address that basic flaw ... that of the discontinuation of continuity (and thus of the applicability of standard mathematical solutions as provided on this and the main page).

Your theories do not fit the facts. That is fertile ground for all manner of modern superstitions that are counterproductive to advancing the human race.

Steaphen (talk) 22:58, 20 June 2008 (UTC)Steaphen

Actually, what I propose simply reinstates time into Zeno's arrow "paradox". Zeno's arrow paradox is the t=0 projection of space-time. I merely illuminated the complementary d=0 projection of space-time and then re-integrated the two projections. The reason Zeno's arrow paradox looks like a paradox in the first place is that Zeno strips away the time dimension with a sleight of hand - he relies on imprecise colloquial language to hide the fact that an infinite sequence of t=0 "now"s can only exist within time and cannot possibly constitute time. That is, Zeno shows you a t=0 projection of space-time and rightly asserts that motion is nonsensical (in such a projection), but he weasel-words things to lead you to believe that he's talking about normal space-time. Again, I just called his bluff and shined a light on the complementary d=0 projection of space-time.

The calculus, on the other hand, doesn't solve Zeno's arrow paradox - it's just not relevant. Calculus deals with limits, and so can only approach Zeno's t=0 case; calculus looks at things as t tends toward 0, and so space-time remains fully intact (there is no t=0 projection).

That said, I see that you assert that Zeno's paradoxes "MUST inevitably pass through quantum scale increments". Okay, fine. But let's also illuminate the fact that many of Zeno's paradoxes pass through macro-scale increments, as well. The implication then is that quantum mechanics is relevant to Zeno's paradoxes at best only a small fraction of the time (the small fraction associated with quantum-scale increments). But again, I don't think you need to look beyond Zeno's sleight of hand in order to resolve the arrow paradox.

On the other hand, that very small fraction of relevance may be of critical importance to Zeno's dichotomy paradox. A space-time that is infinitely divisible by two into smaller and smaller halves is the mathematical model that Zeno shows us in order to "demonstrate" that motion is impossible. But this time he's begging the question. A more scientific way of looking at this is to start by asking the question, "Is space-time continuous?" Zeno's paradox can be seen as a counter-example, and thus by reductio ad absurdum space-time cannot be continuous. A smallest (quantum) increment is then potentially where the continuous-space-time mathematical model "bottoms out". (Much like how a fractal-geometry model of Britain's coastline "bottoms out" at a grain of sand.)—Preceding unsigned comment added by 97.73.64.154 (talk) 04:56, 24 June 2008 (UTC)

I was intrigued to note that your original reply (sans the last paragraph above) was eventually augmented with an astute observation concerning fractals.

Fractals, in broad terms, reflect various interfaces between the infinite and the finite - which of course is highly relevant to resolving Zeno's Paradoxes.

It seems to me that your next step would be to question what might lay between the various "grains" of physicality.

Steaphen Pirie (talk) 05:06, 11 January 2009 (UTC)

Director

Belief Institute

see also "Congruent Solutions to Zeno's Paradoxes"

Further to your excellent analogy of "grains" of space-time, is this extract from a recent NewScientist article (universe-as-hologram) -- :

"According to Craig Hogan, a physicist at the Fermilab particle physics lab in Batavia, Illinois, GEO600 has stumbled upon the fundamental limit of space-time - the point where space-time stops behaving like the smooth continuum Einstein described and instead dissolves into "grains", just as a newspaper photograph dissolves into dots as you zoom in."

Steaphen (talk) 10:11, 17 January 2009 (UTC)

which echos similar commentary from around 10 years ago...

"We know now, however, that it is Einstein's theory that ultimately fails. On extremely fine scales, space-time, and thus reality itself, becomes grainy and discontinuous, like a badly overmagnified newspaper photograph. The equations of general relativity simply can't handle such a situation, where the laws of cause and effect break down, and particles jump from point A to B without going through the space in between."Steaphen (talk) 06:24, 20 February 2009 (UTC)

I don't get it

From the article: "Yet another proposed solution, that of Peter Lynds, is to question the assumption that moving objects have exact positions at an instant and that their motion can be meaningfully dissected this way. If this assumption is challenged, motion remains continuous and the paradoxes are avoided"

I don't understand how motion remains "continuous" if objects no longer have exact positions at an instant. I can make sense of the notion of "continuous motion" if objects do have exact positions at exact moments: continuous motion would then be that the objest goes through positions/moments that can be quantified by locations on a real number line, or at least that between any two positions/moments there is another position/moment. But when the requirement of objects having exact positions in space or time is dropped, I can no longer make sense of the notion of "continuous motion". But, of course, just because I can't, doesn't mean that there isn't a way to make sense of this. Can anyone make sense of this? —Preceding unsigned comment added by 128.113.89.96 (talk) 15:13, 23 August 2008 (UTC)

A Solution?

These arguments don't work because things have thickness. One object will catch up with the other object once the distances you keep dividing in half get less than or equal to their thickness. Or am I saying something stupid? —Preceding unsigned comment added by 71.135.44.251 (talk) 03:49, 21 September 2008 (UTC)

I don't think you're saying anything stupid. Since Zeno's argument treats the objects as points, and since actual objects have thickness, indeed a little more work needs to be done in order for Zeno's argument to apply to the motion of real objects. However, one relatively easy thing to do would be to consider (at the various points in time) the point in space of the object that is closest to the destination. If you do that, then Zeno can still point to the existence of an infinite sequence, and from there his argument can proceed as normal. So, I don't think that the fact that real objects have thickness solves anything. —Preceding unsigned comment added by 128.113.89.96 (talk) 17:26, 25 September 2008 (UTC)

A related thought.

Perhaps whether or not there are infinite points along the path is irrelevant. A point in space is just that, it does not itself have mass (and so we can easily wrap out minds around there being an infinite number of them) or move (rather, objects move relative to these points). But even if we think of a moving object as not having mass (ie, not having thickness), there could very well be a limit to the smallest distance that objects can move (like the Planck length).

If that were the case, the "object will catch up with the other object" when the "halfway" distance is less than or equal to the absolute minimum distance that an object (even one without thickness) can move. The infinite sequence relies on the assumption that this minimum distance will always be less than the next halfway length, or that such a minimum does not exist. —Preceding unsigned comment added by 68.8.181.178 (talk) 08:09, 16 July 2009 (UTC)

A question?

This is more of a question than anything else, but given that in general the simplest solution to a problem is to be preferred to the more complex one, could it not be the case that Zeno (and Parmenides) were in fact right, and that motion is (simply) an illusion? (If that's wrong, why is it wrong?) Thanks! —Preceding unsigned comment added by 79.97.236.37 (talk) 05:24, 30 December 2008 (UTC)

I think time is the illusion. I am actually intrigued by what a lot of spiritual teachers are saying, that time does not exist and that it is really just a mental construct. I actually read a pretty interesting argument about time on youtube (surprise!) where two guys were discussing how cause and effect is impossible because for one "event" to affect another, they would have to exist in the same, exact moment in time. But if time exists, then each are in different moments in time (however finely divided..), so how can they affect one another? And even if time could be divided into infinitely small segments, then just like the argument 0.999... = 1, we could say that time would never pass (because each segment basically approaches zero). And if time could be broken down into discrete lengths, then the "jumps" between different moments wouldn't make sense, and cause and effect would be broken because each event is fundamentally separated from the other. When you observe reality, you can find no *direct* evidence that time exists, there is only a continuous present moment (I don't know what this means in terms of allowing for motion either, though).

So basically, I think there is a whole new order of reality that we don't understand yet (a greater reality), or are somehow oblivious to, but that currently the "theory" that "time" exists is impossible and as far as I see it, basically just a mental construct (i.e. you never experience past or future) 142.150.72.132 (talk) 16:31, 15 January 2009 (UTC)

Quantum zeno effect

We should discuss the pros and cons of including this section (rather than edit warring over it). Any thoughts? hgilbert (talk) 17:06, 4 March 2009 (UTC)

Good idea. I think the section should go, because I think it isn't related to Zeno's paradox. Zeno's paradox is fundamentally about infinite sequences, whereas the Quantum Zeno effect is not. —Preceding unsigned comment added by 67.244.167.93 (talk) 03:02, 13 March 2009 (UTC)

Why resort to calculus or infinite series?

The Achilles/Tortoise problem is exactly the sort of thing I would expect to see on a Physics I exam, and the solution can be found using algebra and not calculus or infinite series. I don't think this simple solution is adequately described in the article even though the results are mentioned. A more detailed explanation is below, but maybe it could be shortened for the article.

Using the numbers currently in the article we can form two equations. One equation for the X position of the tortoise (Xt) and the other for the X position of Achilles (Xa). Both equations are with respect to time, t.

The tortoise starts out 100 feet ahead and runs at 10 feet per minute: Xt = 100 + 10*t

Achilles runs at 100 feet per minute: Xa = 100*t

Assuming we are neglecting time dilation due to their different speeds and the issues with simultaneity these two share the exact same time, t. We want to know how long it will take for Achilles to reach the tortoise, when Xt = Xa. Setting the two equations equal we get:

100 + 10*t = 100*t
100 = 90*t
t = 100/90 = 10/9 = 1 + 1/9 = 1.11111... minutes

All we would have to do to make Achilles arrive at the tortoise at an even 2 minutes is set the tortoise's lead to 180 feet instead of 100. Doing this may make the numbers a little more clear since some people may be confused by the infinitely repeating decimal. I would have added this explanation to the article myself, but I figured I should check the talk page first.

If you graph the two equations it's plain to see the point where Achilles overtakes the tortoise. You can pick as many points along those lines as you want and call each segment a task, but it doesn't change the fact that the lines intersect. We know that Achilles moves at 100 feet per minute, this 100 feet can be divided into infinitely many sections but he will still traverse it in 1 minute. Using the given data the solution is straightforward. The only way this solution can be false is if one of the givens are false, in which case this problem is irrelevant. —Preceding unsigned comment added by 97.104.127.235 (talk) 00:01, 15 April 2009 (UTC)

Hi. The third paragraph under Proposed Solutions in the main article provides the kind of calculus based solution you are proposing. This is indeed the most common way in which people react to Zeno's paradox. But the article also points out a problem with this kind of solution, which is that it really doesn't get at the heart of the paradox. To be specific, calculus will tell us where and when Achilles will overtake the Tortoise, but it doesn't explain how this point in space and time can ever be reached, and it was the latter that Zeno as concerned about. Put another way, calculus gives us the exact time and location of Achilles overtaking the Tortoise if such an event were to take place, but the problem is that Zeno gave us an argument that concludes that this event cannot take place. Now, of course, all this sounds pretty ridiculous, as we know that this event will take place! But please understand that that is exactly what makes this a paradox, i.e. how is it possible to have a logical sounding argument with such an obviously false conclusion?!? And the answer to that is that there must be something wrong with the argument, either in its logic, or in its assumptions. So, any resolution to the paradox will have to figure out exactly where Zeno's reasoning is going wrong. A solution has to point to some assumption or to some inference that is mistaken. And unfortunately, it doesn't look like calculus points to any such flaw in Zeno's argument. It merely provides us with a separate line of reasoning (a calculation, really), that verifies that Achilles will overtake the Tortoise. But we already knew that. And, much more importantly, it doesn't question any of the inferences or assumptions made by Zeno. So, it doesn't solve anything. But to make things more complicated, there is in fact a version of Zeno's argument you will often hear, and in which there is an obvious flaw, and where calculus can be used to point out that flaw. This is where near the end of the argument, it says something like "and because Achilles needs to do infinitely many things, each of which takes some finite amount of time, it will take him an infinite amount of time to do these things, and so he will never overtake the Tortoise". Now in this line of reasoning is an obvious flaw, as it assumes that the sum of an infinite number of terms needs to be infinite, which calculus can easily show to be incorrect. And I have to believe that this is why so many people believe calculus provides a solution to the paradox. But, this version of the paradox misrepresents Zeno's argument. Zeno would simply proceed at the previous point with "and because Achilles needs to do infinitely many things in order to overtake the Tortoise, he cannot, because you simply cannot reach the end of an infinite sequence; by definition, you can't finish doing infinitely many things (sequentially that is, but that is what we are indeed dealing with in Zeno's paradox)". Of course, at this point people will say: "Wait a minute, but you can do an infinite number of things, simply do each thing at half the time you did the previous thing. Again, calculus will then show that if you do the first thing at t = 1, then you will have done an infinite number of things at t = 2". But again this response doesn't work, because it assumes that time is able to progress to t=2 even as there are an infinite number of points of time t = 1, t = 1.5, t = 1.75, for time to go through, i.e. you would have to assume that an infinite sequence can be finished in order to show that an infinite sequence can be finished. So this is a circular argument. But this also explains why some other people propose as a solution to Zeno's paradox that not all of these mathematical points in time have a counterpart in real life, i.e. that time (and space) are not continuous, as that would avoid getting into infinite sequences in the first place. —Preceding unsigned comment added by 67.244.167.93 (talk) 12:41, 15 April 2009 (UTC)

Fascinating to watch Galilean Priest Sydnrome at work here (Douglas Adams explained it wonderfully with his Somebody Else's Problem). Okay, some suggestions:

1. Begin with evidence from quantum physics (e.g. little things, ipso facto big things like spaceships, cricket pitches, arrows & hares etc jumping around without travelling the space in between).

2. Surmise, reasonably, that space-time is not infinitely divisible (since quantum theory indicates otherwise, and besides, uhm, like where's the physical evidence to show that it is infinitely divisible, aka perfectly geometrically continuous). Okay now the fun bits...

3. Imagine (if you will) a dimension that is, and will remain "immathematical" (ie. the infinite). Next, since our physical reality appears fairly stable and solid, assume it must somehow be 'cycling' into actuality exceptionally fast, like ~10**whatever times per second to give the comforting illusion of stability and order. The actual figure is not important. In other words, surmise some process cycling us through immathematical -> mathematical, back into immathematical. Now, since we're obviously doing this 'cycling' along with everything else, what do we notice about our everyday experiences that might suggest some mechanism for this process? Hint: where do thoughts come from? Brain. Wrong (mostly), since thoughts require firing of neurons, which require rising potentials & collapsing of wave-functions -> waves -> future ("everything in the future is waves"/Sir Bragg). Physical things e.g. neurons can't be causing prephysical possibilities to occur and to 'congeal' into whatever we're currently thinking ... such as what you thinking about this post. What's really causing you to think what you think? How do we tap and work prephysical possibilities? 2nd hint: check out Dr Damasio's work.

While I'm being fairly flippant here, it behoves all of us to reflect on what results we're inviting into reality by sticking with mechanistic cause-effect beliefs. So many wonderful potentials being wasted, ignored and denied through allegiance to outmoded belief-systems. We're capable of far more than what we think and we do ourselves (and others) an immense disservice by expecting that we can entirely count/calculate that which will remain uncountable (as Einstein remarked, not everything that counts can be counted, not everything that's countable, counts).

You're able to follow my comments in more detail at twitter.com/beliefdoctor or at The Belief Doctor

With good wishes, Steaphen (talk) 05:07, 19 April 2009 (UTC)

This problem essentially reduces to "is motion possible" since any motion can be thought of an infinite series of smaller motions. If we are given that there is motion then it follows that infinite series are completable. The point I tried to make earlier (towards the end of the first post in this section) is that if we are given that there is motion then there is not problem, if we are not given that motion is possible then that is the problem we need to solve and we can forget the tortoise stuff. If motion is not given as possible then I think the rest of the problem is just fluff. 97.104.152.15 (talk) 23:26, 27 April 2009 (UTC)

Well, there is certainly a large group of people who regard Zeno's paradox as one about the possibility of motion. The paradox according to these people is that Zeno gives us an argument against the possibility of motion, while at the same time motion certainly seems possible. To resolve that paradox, you either have to agree that apparently motion isn't possible after all, or you need to find a flaw in Zeno's reasoning.

However, a second group of people don't regard Zeno's paradox as a question of "is motion possible", but rather as a question of "how is motion possible". And now the paradox is that while many people think of motion as an infinite series of smaller motions (you seem to be one of those), Zeno argues that under such a view of motion, motion becomes impossible (which, by the way, goes directly against your claim that if motion is possible then infinite series can be completed), and since motion is possible (on this view, the possibility of motion is indeed a given), this view of motion is incorrect (and so motion is a given, but we also have a problem, which goes directly against your claim that if motion is a given, then there is not a problem). In short, the paradox is while we see motion as an infinite series, Zeno argues that that cannot be what motion is like. To resolve this paradox, you either need to agree that apparently motion is not an infinite series of smaller motions (for example, maybe space and time are not continuous), or you need to find a flaw in Zeno's reasoning.

Well put. However, I would suggest going further, by accepting the enormous successes of calculus, and standard geometric approaches as pointing towards validity within some deeper, or expanded context. In other words, we accept the power of infinite series, but we don't necessarily assume they apply exclusively to our physical system. The enormous, unparalleled success of quantum theory rests on the 'and' operative (e.g. Feynman's Sum Over Histories approach, reliant on summations of history A and history B and ... all alternative paths, not one-track Newtonian mechanics).

With an expanded understanding, we can recognise the error in attempting to undermine Zeno. He was right, motion is impossible if one assumes perfect continuity of our physical space-time. If that weren't the case we could perfectly track and predict itty-bits of stuff, without any problems. Plus we wouldn't be seeing the experimental evidence of quantum mechanics. In the end, Zeno wouldn't have had a leg to stand on. (Please excuse the pun)

I think the reason for the clear examples of GPS (above) is that without that expanded framework of understanding, many will hang on to their limited (one-track, one history) world view, despite consistent evidence that their belief-system is no longer congruent with reality.

The worthwhile question then becomes, how do those who are constrained by their beliefs, step into a new abyss, one that cannot be entirely reasoned, or analyzed, and do so without going insane (as many inventors, creative scientists, artists, writers, entrepreneurs, leaders and deep thinkers have done, and do). This is not a trivial consideration or question. Those who remain locked within the old Newtonian one-track, one-past belief system (and who rest on the validity of a mechanical world-view) are particularly in danger ... as exampled by the experiences of Cantor.

This is not to suggest that all artists, writers, creative entrepreneurs and leaders navigate the abyss with ease and impunity. Merely that they at least have sufficient faith to step and explore, rather than resting on top (not even near the edge) criticizing those who are in amongst it, feeling their way, hoping to report back their discoveries for the benefit of one and all.

Yes, some (artists) go too far, too deep and stay too long in that great chaos, and don't altogether make it back, but they at least acted on their intuitions.

Actually I wasn't saying that I believe motion involves an infinite series of smaller motions, but that was brought up during this discussion and I wanted to address it. Someone argued against my earlier post using that idea. I wouldn't be all that surprised if space-time was quantized.

The poster immediately above me mentioned quantum mechanics, I don't have a thorough enough understanding of quantum physics but perhaps part of the answer to the paradox lies in the fact that at the quantum level determinism disappears and the locations of particles becomes probabilistic. Particles don't necessarily need to travel the distance in between two locations, there is a certain probability that they will simply appear in the new location. The scale at which particles jump instead of travel normally could be seen as the point where the "infinite series of smaller motions" idea breaks down. 97.100.232.24 (talk) 20:09, 29 April 2009 (UTC)

Quantized spacetime, infinite sequences, and the state of this article

See and feel free to contribute to the extensive discussion of this important topic at /Quantized spacetime. The discussion has unfortunately grown too long for the main talk page; please continue it on the sub-topic page above. hgilbert (talk) 11:12, 11 July 2009 (UTC)

Menus, maps and mathematics

For those interested in a condensed version of the discussion in the above section, we may boil down the issue at hand as a failure to recognise the natural reduction of dimensionality involved when attempting to explain, portray or describe any phenomena or experience.

For example, a motion picture film on a screen is a 2-dimensional representation that might portray the general flow of actual physical events, such as a car-chase. But viewing the film can not substitute the rich, sensate experience of actually being in that car-chase. The sights, smells, forces and accelerations of real life are only vaguely hinted at in any film.

Similarly, our lived 3 or 4-dimensional physicality is a reduction of a deeper multidimensional meta-physicality, which is only hinted at in our daily physical lives. Artists, writers, sages and others who succeed in feeling this reality intuitively know that our physicality is only a limited reflection of deeper meta-physical dynamics and potentials.

Much mention is made of infinite series in regards to solving Zeno's Paradoxes. These series echo a deeper base or ground from which only limited "frames" (pops, pulses, coagulations, choices and experiences) occur in physicality.

The efficacy of calculus and mathematics is not the issue ... the issue at hand is to realise that they (the infinite series solutions) echo or reveal how our limited-frame world is, and will continue to be an on-going reduction or congealment of a far richer, multidimensional meta-physical existence. The experimental results of quantum physics reveal a limited, or finite-framed actuality reduced (or "collapsed") from infinite possibilities.

To suggest that we tangibly (in literal terms) experience this infinite-dimensionality is to suggest that watching a film is the same as living the experience portrayed by the film.

Just as a film of other lifestyles or circumstances can hint of possibilities that we ourselves might wish to live, so too do the mathematical solutions hint of the underlying possibilities and potentials not yet physically experienced.

Good wishes
Steaphen (talk) 09:46, 7 July 2009 (UTC)
Belief Institute

Infinity

There is absolutly no proof that a infinate sequence of points exist, one could argue that for an object to get from A to B it would have to pass an infinate amount of points, and therefore it would require infinate time. This is wrong however, consider an object leaving A, as it changes from from motionless to moving it must reach the first point. Likewise when it arrives at B it changes from moving to motionless and therefore must reach the last point. The infinate number of points does not exist in reality, just in the mind 193.120.116.183 (talk) 00:23, 28 May 2009 (UTC) shane mc donnell

That reality contains infinitely many places or instants has never been disproved, either. First or last points in the sense you use them do not exist, and arguments for the impossibility of changing between motion and rest are in the article. Regards, Paradoctor (talk) 08:19, 28 May 2009 (UTC)

Infinity . . .

Does the fact that it has never been disproved make it any less probable? If so, one could argue for the existence of lepracauns, goblins and fairies also! The arrow paradox only works if you remove the element of time, the idea of motion being impossible without time is no paradox, simply common sense. The clever wordplap may lead you to think the time has simply reduced, when in fact it has been removed completly 193.120.116.147 (talk) 19:43, 29 May 2009 (UTC) shane mc donnell

Anon, I don't know how long you've been around, but there are a lot of rules on Misplaced Pages. These may seem daunting at first, but you can always ask for help, and the better you get to know the rules, the better your editing experience will be. You may want to start here. The statements in your above comment do not appear to come from any reliable source, and thus should not be discussed in here. If they do come from a reliabe source, please tell us from which one, so we can start discussing how to integrate them into the article. Regards, Paradoctor (talk) 21:35, 29 May 2009 (UTC)

Not seeing the trees for the forest

It seems many have difficulty understanding the essential dilemma posed by the advances in quantum theory in relation to Zeno's Paradoxes. Some respondents at this website have even suggested that quantum physics has “nothing” to do Zeno's Paradoxes.

Zeno's observations concerned the movement of physical things such as arrows, runners and the like.

The use of Newton's or Leibniz's calculus (based on infinite series) was and is used to great effect to track and predict the movement of everyday objects, such as an arrow as it travels along its trajectory.

The efficacy of such tools (differential/integral calculus) is taken to mean that such tools are entirely sufficient for explaining the full and complete nature of physical movement.

This appears to be the reason for the almost universal use of standard Newtonian solutions (involving infinite series) on this and similar websites.

However, a thought experiment was previously introduced (on the “Archive Page”) that calls into question this assumption. The thought-experiment (with some minor amendments) is repeated here for clarity and convenience:

From the section Zeno's Paradoxes, Wikpedia and Apple Carts

Part A:

The arrow moves through physical space in a trajectory in accord with Newton's laws (e.g. in a parabolic curved trajectory). At each point along that trajectory, mathematically we can say at point 'x' along its path, the arrow will be at 'y' height. Moreover, at any point along its path we can mathematically determine its physical characteristics of position, momentum and rate of acceleration (given known initial conditions).

The variables in describing the flight of an arrow all have physical attributes (e.g. momentum, position, rate of acceleration etc.).

Let’s imagine (a thought-experiment) in which we fashion ourselves a particularly fine arrow (one so fine that its tip has been honed to just one iron atom). Since, according to classical physics we can accurately determine the arrow's trajectory, we can likewise accurately determine the position of the atom.

Lets imagine we fire the arrow in a complete vacuum, so as to not to confuse the issue with friction losses, cross winds and other physical influences.

According to classical physics, at any time during its flight (assuming known initial velocity, weight of the arrow) we can apply classical equations to physically predict not only the arrow’s (and the atom’s) position, but also its momentum.

Note: we were particularly diligent and analysed the arrow to determine not only its weight but also the number of atoms and types of atoms it contained (with their respective atomic masses), thus enabling momentum for the lead atom to be accurately determined.

When geometric series are used to “resolve” Zeno’s Paradoxes, the assumption is made of perfect continuity – that is, there are infinite points along its path.

Accordingly, at any point in time, we can precisely determine the position of the arrow’s lead atom, and its momentum. Unless there is something wrong with my reasoning, this means we’ve busted the Uncertainty Principle of quantum theory. Nobel Prize please.

This thought-experiment is used to demonstrate how standard mathematical solutions that rely on infinite series are not relevant or applicable to the finer movement of physical things.

Quantum theory and experiment exposes the error of such treatments. Quantum theory has a pivotal role (at least until superseded or eclipsed by a better theory) in regards to solving Zeno's Paradoxes.

To suggest otherwise by quoting or considering solutions involving infinite series, which necessitates avoiding the reality of the experimental evidence of quantum physics, is to offer theories without a modicum of clarity of reasoning or observation.Steaphen (talk) 09:37, 31 May 2009 (UTC)

Clarity of cause

In reviewing the intransigence over accepting the idea that Zeno's Paradoxes cannot be solved using infinite series (see the above section "Not seeing the trees for the forest") I began to reflect on the deeper issue at play.

I believe the core of the issue is one of immaturity and short-term expediency.

Specifically, when we believe in a mechanical, objective universe, we are able to disconnect, or distance ourselves from intimate association with the resulting system. We are, in effect, able to lay blame elsewhere, claiming that we are victims in a random, senseless universe, or at the mercy of a vengeful God (science and religion, resp.).

A mature self-organising (holodynamic) systems world-view carries with it the responsibility for the reality we share.

To quote Freeman Dysan, Emeritus Professor of Physics, Princeton.

Quantum mechanics makes matter even in the smallest pieces into an active agent, and I think that is something very fundamental. Every particle in the universe is an active agent making choices between random processes.

...consciousness is not just a passive epiphenomenon carried along by the chemical events in our brains, but is an active agent forcing the molecular complexes to make choices between one quantum state and another. In other words, mind is already inherent in every electron.

As within the part, so within the whole. As within the micro, so within the macro.

Zeno's Paradoxes telegraph that the accepted objective view of our physical universe is untenable. The world we experience, one amongst limitless others, is intimately connected with, and co-created through our choices.

Objective science is a limited snapshot of a far deeper multidimensional existence. To say or believe otherwise requires a small-ego persona that seeks to remain disconnected in order to avoid responsibility for the resultant reality.

While appearing expedient in the short term (e.g. enabling fast food production using antibiotics/ growth hormones / caged hens / testing medications & cosmetics on animals etc) we ultimately debase ourselves, deepening our disconnect with the natural world.

That, I sense, is the reason many will continue to avoid recognising why we cannot solve Zeno’s Paradoxes using infinite series.

Steaphen Pirie
Director
Belief Institute —Preceding unsigned comment added by Steaphen (talk • contribs) 15:12, 2 July 2009 (UTC)

The ultimate issue in Dichotomy Paradox - Walking the Planck

Eventually the distance in the The Dichotomy Paradox will equal the Planck Length. For the moment, I will assume there is no length shorter than the Planck Length. When the distance reaches this lenght, a decision must be made. Round up, or Round down. Zeno might be happy to know that rounding down would mean not moving. However, rounding up is kind of cheating, since the distance is literally divided by nothing (note, this is point where nothing reveals itself to be not equal to 0. To bad the Greek philosophers didn't spot that one earlier). Technically, you can't divide it ad infinitum. Thus you cannot fulfill the conditions of the paradox. Thus the paradox is not valid. But wait, by Walking the Planck or more, which everybody capable of walking can do, don't you have to pass half the length of the Planck Length before passing a full Planck Length? Well, nothing is smaller than a Planck Length so obviously not. Zeno might have a point about movement. Suppose moving from one Planck Length to the next is not accomplished by movement, but rather the object simply is in another place than where it was. This is known to happen in quantum physics (unless QP is completely baseless). But wait, its more complicated than that. Light traverses 1 Planck Length in a 1 Planck time. If Homer is not traveling at least the speed of light, then its impossible for him to move at all, because he'd travel less than a Planck Length. Traveling less than a Planck Length is not possible so Homer must not be moving when he appears to move or is moving at least the speed of light. Maybe there are no Planck units. Or maybe I'm just to tired to be reading about Zeno and responding at this time --Zerothis (talk) 06:17, 13 July 2009 (UTC)

Looks good to me :) Particularly your conclusion, "Suppose moving from one Planck Length to the next is not accomplished by movement, but rather the object simply is in another place than where it was." - that sounds suspiciously like a description for quantum superpositions. Which, when applied to everyday life, means as we step, there's two of us, one of the front foot, one on the back. But hey, maybe some prefer to stay on the back foot, destined to repeat what's been done, what's been said, what's been written, never managing another step forward. Ciao Steaphen (talk) 12:27, 14 July 2009 (UTC)

For those interested, Microsoft has made Richard Feynman's lectures available online.Steaphen (talk) 22:34, 16 July 2009 (UTC)

From Lecture 7, Seeking New Laws:

"...that space is continuous is, I believe, wrong. Because we get these infinities and other difficulties ...I rather suspect that the simple ideas of geometry extended down into infinitely small space is wrong."

- Professor Richard Feynman

The Messenger Series: Seeking New Laws

Many-worlds QM does not require discrete spacetime or the invocation of Planck lengths. mike4ty4 (talk) 20:55, 17 July 2009 (UTC)

Also, discrete motion would not require such a "bifurcation" process either. If you read what you just quoted, they say : "the object simply is in another place than where it was." Nothing bifurcates, nothing "superposes", at one discrete point of time, the thing occupies one discrete unit of space, then in the subsequent one the thing occupies the next. I.e. continuous-vs-discrete spacetime is irrelevant to the question of existence of quantum superpositions. mike4ty4 (talk) 20:59, 17 July 2009 (UTC)

There was no explicit statement suggesting Many-worlds required discrete spacetime. As offered elsewhere you appear to be unable to do rudimentary analysis (e.g. of statements by others). The vast majority, if not all of your posts are about what isn't needed or proved, not about what is physically occuring. You have written of step functions without any connection with, or direct relationship to what IS actually occuring in physicality. Here, in this fine posting by someone who has at least attempted to suggest what IS happening, you again go the negative with what ISN'T needed. This page is about the dilemma of understanding the movement of physical things. It is not, contrary to what many might wish, about the mathematics, although mathematical descriptions might help some.

The point of the step functions and Dirichlet functions argument, and I don't see why you keep missing this, is to try show you that motion exhibiting "jumps" can be accomodated in both a discrete and a continuous space-time, as they are examples of discontinuous functions defined on continua. If you took one as "position" as a function of "time", then it would describe a motion with a discontinuity. That is, its point is to invalidate your claim that such "jumping" motion (which by the way you have not yet provided evidence for -- unless you want to call misunderstandings and misinterpretations of QM theory "evidence". Please, for the gazillionth time, tell me what on Earth is this experiment where one can actually observe the particle "jumping"??? How did they manage to observe the particle in-flight to determine its jumping? And so on...) is proof or evidence of discrete space-time. It is not. What I am doing overall is not just saying "oh, that isn't needed". What I am doing is saying "that argument/evidence doesn't support your claim. Do you have a better one?". mike4ty4 (talk) 07:41, 26 July 2009 (UTC)

Once again, you appear to be unable to perform rudimentary analysis. Any claim as to what a thing is NOT, involves an implicit comparison to what it IS (try thinking complete and utter "nothingness", if you have difficulty understanding this). All of your posts implicitly involve your theories about what reality is. I've simply asked that you be more explicit (and honest) in your descriptions and theories.

As for your "what is this experiment?" confuses me ... it's a nonsensical question: The root of Quantum Theory (Latin root "quantas" for "how much") is about measurement of, and theories relating to discrete lumps and jumps (no one has ever measured a continuous wave, or seen one, they've only ever, and will only ever measure and observe discrete, discontinuous entities, or quanta, that when observed together appear to form waves. If you have difficulty understanding my point, I suggest you go off and search for a forest without trees, and report back when you have found one).

In view of the above, the question that therefore needs to be asked, and answered is: What experiment or physical evidence do you have that supports your theories of absolute, endless continuity of physical movement, and of space-time? Steaphen (talk) 22:19, 8 August 2009 (UTC)

I think that for credibility, if you continue in the negative without substantive ideas of what is PHYSICALLY occuring, mediation will need to be requested. Again, if you wish to offer ideas and theories to explain the experimental evidence, well and good. Otherwise, I think it best you discontinue your negative statements concerning other postings that at least offer some ideas as to what IS physically occuring.

That's because that has not been the point. The point is not to propose a new theory. You have not provided this experimental evidence yet. If you can, I'd like to see it. Because, say, the 2-slit experiment is totally consistent with ordinary QM. These are the things that led to the development of QM! So what is this experimental evidence? Tell me right here, and I'll talk about it. Rather, you came in here proposing a new theory, and you offered arguments to try and back it up. I saw flaws in those arguments, and then decided to point them out to you. As for what is occuring, for all the experiments you have so far mentioned, I'll refer you to QM theory. mike4ty4 (talk) 07:41, 26 July 2009 (UTC)

Many, if not all of my references were to existing interpretations of (or combinations of interpretations of) quantum theory, which, as anyone with some rudimentary ability to reason will concur, must relate to the finer movements of physical things, i.e. quantum theory must be relevant to the issue of Zeno's Paradoxes. To suggest otherwise is a disconnect of a theory about reality, with experienced reality. I seem to recall that psychiatrists have a term for such disconnects ...

As the award-winning physicist Dr Fred Alan Wolf explained, "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." If you have ideas or theories of what is actually happening experimentally, and it indicates Dr Wolf's statement is incorrect, then by all means share your thoughts.Steaphen (talk) 23:57, 17 July 2009 (UTC)

I do not see why this must be incorrect, nor why it proves or evidences your theory, nor why it is incompatible with models like QM. mike4ty4 (talk) 07:41, 26 July 2009 (UTC)

The new religion of mathematics

There are some respondents at this site who refer to various mathematical theories in order to account for, or solve the dilemma of physical movement, as was perhaps first queried methodically by Zeno of Elea.

However, as experimental evidence of quantum physics is now revealing, mathematical theories that rely on continuity cannot be directly and continually applied to account for or explain the evidence of physical movement.

Either something is wrong with the evidence which can be repeatedly and independently observed, or the "simple ideas of geometry extended down into infinitely small space is wrong."

To hark back to theories that once worked reasonably well (Newtonian physics), but which no longer fit the evidence, is no different to superstitiously citing some religious text as being the complete and final truth, in the face of glaring evidence to the contrary.

The travesty of, and blind adherence to religious dogma in Galileo's time is again surfacing in our time, as the blind adherence to "simple ideas of geometry".

As was exemplified in Galileo's time, the superstitious adherence to old belief-systems can do immeasurable harm to humanity's advancement and well-being.

Steaphen Pirie
Director
Belief Institute
www.beliefinstitute.com
Steaphen (talk) 09:45, 18 July 2009 (UTC)

In view of the "invitation" to clarify the errors in modern scientific thinking, I've expanded the above materials into two articles/posts on the Belief Institute website:

The Travesty of Modern Science

Moving Beyond a 2,450-year-old era

From the book An Introduction to the Study of Experimental Medicine by renowned French scientist, Dr Claude Bernard:

If a doctor imagined that his reasoning had the value of a mathematician's, he would be utterly in error and would be led into the most unsound conclusions. This is unluckily what has happened and still happens to the men whom I shall call systematizers. These men start, in fact, from an idea which is based more or less on observation, and which they regard as an absolute truth. Then they reason logically and without experimenting, and from deduction to deduction they succeed in building a system which is logical, but which has no sort of scientific reality.

Superficial persons often let themselves be dazzled by this appearance of logic; and discussions worthy of ancient scholasticism are thus sometimes renewed in our day. The excessive faith in reasoning, which leads physiologists to a false simplification of things, comes, on the one hand, from ignorance of the science of which they speak, and, on the other hand, from lack of a feeling for the complexity of natural phenomena. That is why we sometimes see pure mathematicians, with very great minds too, fall into mistakes of this kind; they simplify too much and reason about phenomena as they construct them in their minds, but not as they exist in nature. Steaphen (talk) 07:37, 21 July 2009 (UTC)

Order, please!

Might I remind everyone of WP:SOAP and WP:TALK? Paradoctor (talk) 08:43, 21 July 2009 (UTC)

Thank you for this reminder. It was good to read the list of "Do not"s (I must admit I had not previously read them). I note that one of the "do nots" is "Misplaced Pages is not a collection of unverifiable speculation."

Without getting on my soapbox (another 'do not') I think that policy is quite appropriate for this Discussion page. If it is not verifiable, we should not include reference to it. But that begs the question, in regards to this particular topic (Zeno's Paradoxes), isn't the idea of solving Zeno's Paradoxes entirely speculative, in that no one has ever actually verified moving through an infinite series? In the above section, I included the quote by Claude Bernard because, as he so eloquently puts it, to only rely on logic to explain real life leads to being utterly in error.

Perhaps on that basis, the main article should be rewritten to reflect the speculative nature of the mathematical theories contained therein. Or perhaps on purely technical grounds, references to infinite series in relation to Zeno's Paradoxes should be removed entirely, given that they remain speculative and therefore contrary to Wikipedian good policy.Steaphen (talk) 00:32, 22 July 2009 (UTC)

Paradoctor - in reviewing my posts (above) it would seem I have breached Misplaced Pages policy. On that basis, I'd be happy for you, or whoever would be appropriate to remove all my posts for the sake of adherence to good policy (The key issues and implications covered above are on the Belief Institute website).Steaphen (talk) 14:41, 23 July 2009 (UTC)

I just saw this. I'm going to quit right now (as of this posting) and take this to a different forum if I want to continue it further. So don't bother responding to the posts I already left a few minutes before this specific writing. mike4ty4 (talk) 08:04, 26 July 2009 (UTC)

Proof of the impossibility of movement

1. Scientific Assumption #1: The standard, widely-accepted scientific solution for explaining the paradox of physical movement (often referred to as Zeno's Paradoxes) is fully resolved by the mathematics of infinite series - that we are able to traverse each point in an infinite sequence of small 'infinitesimal' contiguous and continuous steps in finite time, thus enabling everyday movement of our bodies etc.

The mathematics is more like that you can define a continuous function from a continuum to a continuum, but go on... mike4ty4 (talk) 08:00, 26 July 2009 (UTC)

more like ...? what? we're talking reality here. Physical stuff. What are on Earth, are you on about (pun intended).

2. Assume that the common denominator for all experience of physical movement is our personal thoughts and feelings (awareness). As a corollary, events that we believe occur independent of our awareness, cannot be classed as events per se, but as unverified, speculative beliefs about imagined events (since we have not experienced them).

3. Scientific Assumption #2: Neurological activity (namely, firing of neurons in the brain, electrical signalling in the body) is the cause for all our thoughts and feelings (awareness).

Materialism! But OK, if you want... mike4ty4 (talk) 08:00, 26 July 2009 (UTC)

Materialism? Moi? That is the assumption of mainstream, classical science - hence the impossibility of movement - see below (pun intended).

4. Scientific Assumption #3: Physical movement of our bodies directly corresponds to, and results from neurological activity, and that 'subconscious' processes are still the result of neurological activity.

Not yet a problem, at least when not taken as seriously as you attempt to do it NEXT... mike4ty4 (talk) 08:00, 26 July 2009 (UTC)

"Not yet a problem", yet you respond. Why? Your posts will stretch the patience of any visitor to this site.

5. Scientific Assumption #4: Each thought, choice and movement we initiate originates with electrical activity in the brain (namely, with the firing of neurons).

Again not a problem yet... mike4ty4 (talk) 08:00, 26 July 2009 (UTC)

Again, courtesy, please. Consider other visitors to this site.

6. In view of the fact we cannot think the infinite, for whatever we think, we will need to think again, and again ... ad infinitum (thus never fully completing our task of thinking the infinite), there can be no 1:1 correspondence of neurological/thought activity, with each infinitesimal step in movement, since there is no corresponding "infinite firing" of neurons. (In any event, the carriers of electrical signals (electrons) are quanta that, according to quantum theory, do not move continuously, thus do not provide the continuity required by point 1).

And why should there be a 1:1 correlation? See below, this has nothing to do with the fundamental structure of the universe. mike4ty4 (talk) 08:00, 26 July 2009 (UTC)

mike4ty4, you appear to misunderstand the assumptions of materialistic science (e.g. the vast bulk of science). Each infinitesimal sub-Planck movement either has a physical cause, or it does not (i.e. the cause is non-physical). I'll assume you would assert the former, namely each infinitesimal movement has a physical, and physiologic cause. Please explain what that physiological cause is. If no such explanation, then you have confirmed that due to the assumptions as given, movement is impossible.

7. Ipso facto physical movement of our bodies is impossible, since by definition the neurological activity that causes infinite infinitesimal physical movements, has yet to occur, and cannot occur.

That assumes that every portion of the movement down to arbitrarily small scale requires neurologic activity.

CORRECT! That is the assumption of deterministic science. Perfect determinism requires an absolute, unambiguous correspondence between physical effect and some physical, identifiable cause. No exceptions.

This is the problem.

Not at all. It is only a problem for those anchored to old-paradigm deterministic theories, such as infinite series solutions to Zeno's Paradoxes.

To make muscle, for example, move a distance, does not require continuous stimulation.

Analyse what you have written. You are suggesting the physical atoms within a finger that moves, for example, have no identifiable physical cause for their continued, sub-Planck movements. But this makes a mockery of the perfect determinism required by infinite-series solutions. Even the momentum of an atom (that would continue an atom's movement in the absence of resistance, and forces from adjoining atoms in the finger) is a physically identifiable and measurable aspect of reality.

Even a pulse can do it. In any case, nerves sure don't fire at 10 times a second, so their firing rate obviously has nothing at all to do with how quantized space and time are, and also, therefore, with whether or not continuous motion is physically possible. This neurobiological problem does not prove anything about space and time.

Perfect determinism (aka infinite-series solutions to Zeno's Paradoxes), absolutely and totally requires that there is a neurological cause for moving a finger (for example) through sub-Planck level movements.

It has nothing to do with the structure of the universe. Therefore once again you have failed to provide a good argument for your theory! mike4ty4 (talk) 08:00, 26 July 2009 (UTC)

Nonsense. Determinism requires identifiable physical causes for physical effects. For every effect, a physical cause can be determined. The world is one big machine, according to standard scientific assumptions. Even chaos theory is deterministic at core. As for mention of 10**43 times a second, what has that got to do with this proof?Steaphen (talk) 09:51, 3 August 2009 (UTC)

Once again, Scientific Assumption #4 states that movement of muscles is biological and neurological in origin. However, there can be no direct biological/neurological origins for infinitesimal sub-Planck length movements, unless the cause for them is "magical".

This section "Proof of the impossibility of movement" made no reference to space-time, merely the fact that if you rely on a materialism to explain physiological movement, you would need to explain how neurological processes create movement on sub-quantum (sub-Planck length) scales. As for the issue of space-time, this post did not reference the issue, nor is it relevant to this proof. It is simply an exercise in applying standard scientific assumptions, to highlight the error of Assumption no. 1. The proof still stands. One or more of the above Assumptions is in error.

Excerpt Belief Institute

Additional comments and implications:

Further to point 6., point 1 (Scientific Assumption 1) requires infinite infinitesimal movements, including movements that must occur at and below quantum scales. In other words, the standard, widely-accepted infinite series solutions to Zeno's Paradoxes requires that our physiology initiates movements shorter than the Planck length – well beyond any measurable limits of quantum theory, and well beyond any currently proposed or theoretical neurological/physiological/quantum-physical processes.

Finally, Scientific Assumption 1 involves absolute determinism (in that for each point within the infinite sequence of points when traversing any arbitrary length) there exists an associated physicality ('particle' or frames of particles).

Thus, string theories and other theories involving multi-dimensional solutions are disallowed by the strict deterministic requirements of Assumption 1, again confirming that physical movement, based on these assumptions, is impossible. Steaphen (talk) 09:50, 27 July 2009 (UTC)

8. Thus, a runner can never start a race; (and assuming a similar necessary neurological base for animal behaviours and actions) a hare can never catch a tortoise, as Zeno originally posited.

♥

If some of you mathematicians would like to help me reframe the above into more formal-logic language, that would be enjoyed and appreciated. In any event (this is my initial rendering of this proof/updated), I'm well enough pleased with it as is. Thanking and appreciating everyone who's 'motivated' me to fire up a few neurons to prove the impossibility of movement.

• • For edited, latest updates on the above proof, read more at Proof of the impossibility of physical movement

Steaphen (talk) 05:46, 23 July 2009 (UTC)

2nd Proof of the Impossibility of Physical Movement

Proving the impossibility of physical movement, based on the assumptions of modern science

This proof, based on the assumptions of modern science and medicine, reveals how we are unable to move our bodies even for the simplest of tasks, such as blinking an eye, or lifting a finger.

1. A person chooses to begin walking. Neurons fire in his/her head, thus initiating electrical pulses sufficient to cause the relevant muscles to work the process of physical movement.
2. Prior to the first signal being sent, a neuron is required to fire (note, see Scientific Assumptions identified earlier in the above section).
3. Prior to this first neuron firing, sufficient electrical charge (potential difference) must be accumulated in order to fire the neuron - for the signal to leap the synapse.
4. However, the building of electrical potential cannot be caused by the choice of walking itself, as it is the first neuron ("thought"/"feeling") in the chain of electrical signals sent to the muscles.
5. Since, according to the Scientific Assumptions (see above), all physical movement originates with neurological activity (firing of neurons), there cannot be a neurological "first thought" or a "first feeling" associated with the choice to walk, as there is no cause of the building of electrical potential sufficient to enable the required neurological processes of "thinking" and "feeling", and the subsequent experience of walking.
6. Ipso facto, physical movement, based on the Assumptions as defined, is impossible.

Steaphen (talk) 13:15, 27 July 2009 (UTC)

A Planck in the eye of Newton

It seems those who suggest or believe that infinite series offer solutions for Zeno's Paradoxes, forget such solutions are based on a "perfect" determinism" that requires a strict and unambiguous 1:1 correspondence of physical cause for each physical effect (e.g. the effect of physical movement of Zeno's runner, hare, tortoise etc).

This strict correspondence requires a physical, neurological cause for sub-Planck scale movements (which must occur due to the deterministic requirements of infinite-series solutions). To suggest that this is not required, is to suggest physical movements have a nonphysical cause.

To move one's finger, for example, through the required sub-Planck scales, begs the question: "what part of our thinking, or neurological activity is driving parts of our bodies through scales that go well below the theoretical limits of quantum physics?"

Imagine the power of our minds to move bodily atoms through scales of movement not only shorter than the Planck length, but infinitely shorter ... at will, on command and with intent.

Steaphen (talk) 10:29, 3 August 2009 (UTC)

Excerpt: "The Mental Universe"

"One benefit of switching humanity to a correct perception of the world is the resulting joy of discovering the mental nature of the Universe. We have no idea what this mental nature implies, but — the great thing is — it is true.
There is another benefit of seeing the world as quantum mechanical: someone who has learned to accept that nothing exists but observations is far ahead of peers who stumble through physics hoping to find out ‘what things are’. If we can ‘pull a Galileo,’ and get people believing the truth, they will find physics a breeze.
The Universe is immaterial — mental and spiritual. Live, and enjoy."

and from the same article:

and ...

Steaphen (talk) 07:52, 9 August 2009 (UTC)

Achilles and the Tortoise

Anybody else realize that this isn't actually a paradox? The reason Achilles never overtakes the tortoise is because the description of time is f(x)=1/x, which never reaches 0. The story takes smaller and smaller increments of time going into the infinitesimal range of numbers above zero, rather than following the linear trend of time progression. 204.158.149.12 (talk) 15:25, 6 August 2009 (UTC)

The use of mathematical functions such as f(x)=1/x (as applied to physical process of movement) relies on the assumption of continuity (of movement, and space-time). Upon what basis do you make such an assumption (of continuity)? What physical evidence supports this assumption, particularly in regards to physical movement at quantum scales, and at sub-Planck scales (as required by infinite series solutions)?Steaphen (talk) 22:27, 7 August 2009 (UTC)

Proof of the impossibility of physical movement (revised)

• Please DO NOT edit or insert comments inside the points below. Add your comments in the comments section, leaving the points listed below unadulterated.*

This proof, based on the assumptions of modern science and medicine, reveals how we are unable to move our bodies even for the simplest of tasks, such as blinking an eye, or lifting a finger.

• Assume that the common denominator for all experience of physical movement is our personal thoughts and feelings (awareness). As a corollary, events that we believe occur independent of our awareness, cannot be classed as events per se, but as unverified, speculative beliefs about imagined events (since we have not experienced them).

1. Scientific Assumption #1: The standard, widely-accepted scientific solution for explaining the paradox of physical movement (often referred to as Zeno's Paradoxes) is fully resolved by the mathematics of infinite series - that we are able to traverse each point in an infinite sequence of small 'infinitesimal' contiguous and continuous steps in finite time, thus enabling everyday movement of our bodies etc.
2. Scientific Assumption #2: Neurological activity (namely, firing of neurons in the brain, electrical signaling in the body) is the cause for all our thoughts and feelings (awareness).
3. Scientific Assumption #3: Physical movement of our bodies directly corresponds to, and results from neurological activity, and that 'subconscious' processes are still the result of neurological activity. Any movement of bodies, or parts of bodies is controlled by neurological processes that start, continue and stop such movements. That is to say, for each and every physical effect (of physical movement) there is an identiable physical cause.
4. Scientific Assumption #4: Each thought, choice and movement we initiate originates with electrical activity in the brain (namely, with the firing of neurons).
5. In view of the fact we cannot think the infinite, for whatever we think, we will need to think again, and again ... ad infinitum (thus never fully completing our task of thinking the infinite), there can be no 1:1 correspondence of neurological/thought activity with each infinitesimal step in movement, since there is no corresponding "infinite firing" of neurons. (In any event, the carriers of electrical signals (electrons) required for muscular contraction and movement are quanta that, according to quantum theory, do not move continuously, thus do not provide the continuity of signal and muscular movement required by point 1)*.
6. As a corollary of point 5 (above), Scientific Assumption 1 in conjunction with Assumption 3, requires that for each and every one of the infinite infinitesimal movements, including movements that must occur at and below the Planck length, there must be some identifiable physical, neurological cause, That is to say, the standard, widely-accepted infinite series solutions to Zeno's Paradoxes requires that our physiology initiates movements shorter than the Planck length – well beyond any measurable limits of quantum theory, and well beyond any currently proposed or theoretical neurological/physiological/quantum-physical processes. On this basis one or more of the above assumptions must be invalid, or the theoretical limits of quantum theory are incorrect. In addition, and as stated Scientific Assumption 1 involves absolute determinism, in that for each point within the infinite sequence of points when traversing any arbitrary length, there exists an associated physicality (measuable, tangible, physical 'particles' or frames of particles). In effect, movement of one's finger requires that it is composed of not only quantum particles, such as quarks, but also 'sub-quantum' particles - 'super-quarks', and 'super-super-quarks' ... ad infinitum. Thus, string theories and other theories involving multi-dimensional solutions (that extend beyond our tangible 3-dimensional physicality) are disallowed by the strict deterministic requirements of Assumption 1, again confirming that physical movement, when based on these assumptions, is impossible.
7. Ipso facto physical movement of our bodies, when based on these Assumptions, is impossible since the neurological activity that causes infinite infinitesimal sub-Planck-scaled physical movements, has yet to occur, and cannot occur.

(Note, the understanding and explanation for how we move is more fully covered in various articles and courses provided by the Belief Institute)Steaphen (talk) 23:23, 7 August 2009 (UTC)

Comments and rebuttals to the above:

Your comments here ...

seriously Paradoctor (talk) 16:34, 8 August 2009 (UTC)

Paradoctor, as mentioned above, I'm happy for you to remove all those posts that breach the Wikipedian rules, including mine. But it seems to me that based on Wikipedian rules the whole of this and the archived talk pages should be removed.

Perhaps you would like to clarify what Misplaced Pages IS, rather than what it is NOT.

Please confirm which of the above posts (and those of the achived pages) entirely conform to Wikipedian rules.Steaphen (talk) 21:34, 8 August 2009 (UTC)

I agree. It is not up to us to wade through all this grandstanding drivel. Just stop this endless WP:OR --JimWae (talk) 03:37, 9 August 2009 (UTC)

Presumably, your posts (on the archived page) weren't "grandstanding drivel" .. upon what basis is yours not "grandstanding drivel"?

As a courtesy to visitors to this site, and potential contributors, perhaps you should at least cite one post that meets all the Wikipedian criteria.

Steaphen (talk) 07:05, 9 August 2009 (UTC)

This talk page is for improvements to the article - not a blog & not a place to hype your blog--JimWae (talk) 08:26, 9 August 2009 (UTC)

Once again, which of the posts conforms to Misplaced Pages criteria? In regards to your "improvements to the article" - correct, my work is to correct the untenable, unsupportable and erroneous opinions in the article, thus improving it. I think given the intransigence, and associated dogmatic assertions by the majority in this talk section, we should seek mediation by Misplaced Pages mediators.

Steaphen (talk) 08:43, 9 August 2009 (UTC)

It would not matter if ALL previous posts did not conform to Wiki policies. We still are expected to follow those policies. However, here is at least ONE that is abot improving the article-- and there are more in that archive. In order to include content in the article, we need to find reliable sources. Your blog does not count as a WP:RS. There are numerous faults easily found in your "Scientific Assumption" approach - but THIS is not the place for people to discuss YOUR theories--JimWae (talk) 20:25, 9 August 2009 (UTC)

Jim, good to see you occasionally dropping in to keep things in order (and all the while using your actual name!). Unfortunately, you appear to be biased in your criticisms. On the main page there is Original Research content in the "Proposed Solutions" section (e.g. by Lynds etc.). The bias appears to be based on your perception of which articles or references are "reliable sources". In Galileo's time, Galileo and his kind were "unreliable" sources, while the Church's authority and scriptures were deemed "reliable" views of reality. Giordana Bruno and to a far lesser extent Galileo experienced the fiery brunt of not obeying accepted opinion as to what constituted "truth." Then, as now, there's no excuse for accepting the prevailing dogmas, just because the crowd says they're reliable sources. "Peer reviewed" I hear you say ... again, by whom, which crowd, which Church?

I can easily gather a crowd of reliable sources, one of whom includes the late Richard Feynman (who, as quoted above believed that "the simple ideas of geometry extended down into infinitely small space is wrong.") Then there's the physicist Professor R.C.Henry from the John Hopkins University, who says that "If we can ‘pull a Galileo,’ and get people believing the truth, they will find physics a breeze. The Universe is immaterial — mental and spiritual." Not to mention a plethora of others voicing remarkably similar ideas, all of whom highlight the error of standard infinite-series solutions to Zeno's Paradoxes. I suppose because their beliefs and comments don't fit your world-view, they must be unreliable sources.

As for my blog, or reference to the Belief Institute website, again, there are many references to other websites that support various theories. I think the unbiased visitor will concur that you're selective in your criticisms. And as for "numerous faults" .. I've had dialogue with a number of folk (e.g. an emeritus professor of mathematics) and as usual, the counter arguments are all based on the assumption of continuity. Yours no different.

Jim, as much as all this might appear trivial, unfortunately there are quite deleterious consequences coming our way (and already upon us) from continued adherence to old mechanical world-views. In as much as I’m part of this reality, it behoves me to do something about the continued adherence to beliefs that served us well enough in a previous era, but which are now dangerously out of step with needed awareness and behaviours.

I set up the Belief Institute as a repository and centre for congruent world-views that go beyond old dogmatic, deterministic beliefs. While my manner is perhaps not as eloquent or as courteous to ensure the greatest acceptance of the message, I am willing to be a messenger in the face of naysaying, abuse and denials.

Steaphen (talk) 08:12, 10 August 2009 (UTC)

Errors on the main page

Jim, in regards to your comments of "improving the article", there are a number of errors on the main page.

Specifically, the comment "Using ordinary mathematics we can calculate both the time and place where Achilles overtakes the tortoise." is incorrect.

We now know that, with the benefit of quantum theory, we cannot precisely do as claimed - we cannot precisely calculate both the time and place where Achilles overtakes the tortoise. The sentence needs correction.

>>UpdateSteaphen (talk) 23:16, 10 August 2009 (UTC): The above sentence has been changed to "Using ordinary mathematics we can approximate (to the limits of quantum theory) the time and place where Achilles overtakes the tortoise."

For similar reasons the statement "More modern solutions using calculus have generally satisfied mathematicians and engineers" is also technically incorrect, as there is no confirmed "solution" to the paradoxes using calculus. Assumptions yes, but solutions, no.

The sentence should instead read "More modern methods using calculus have generally satisfied mathematicians and engineers." (note, change has now been committed).

I'll edit other errors, and add more appropriate content in due course.

Additional errors that need correcting: "The paradoxes certainly pose no practical difficulties." This is incorrect, the quantum paradox (the uncertainty principle - the paradox being that a "thing" can be both entirely uncertain and certain at the same time) presents "difficulties" for those working on microchips, quantum computers and other devices that operate at such scales. —Preceding unsigned comment added by Steaphen (talk • contribs) 23:21, 10 August 2009 (UTC)

Cheers, Steaphen (talk) 18:04, 10 August 2009 (UTC)

You have not established that it is an error to say "we can calculate the time and place". We CAN calculate both. Your *claim* that it cannot be done precsely (whatever that means) is at least open to question. Saying "Using ordinary mathematics we can approximate (to the limits of quantum theory) the time and place where Achilles overtakes the tortoise" is needlessly opaque to the reader. I thus have reverted --JimWae (talk) 23:21, 10 August 2009 (UTC)

We CAN calculate both - whether it is "precise" (whatever that means) is a separate issue. --JimWae (talk) 23:24, 10 August 2009 (UTC)

Whether "Euclidean" mathematics is an accurate model of "reality" (whatever that means) is a separate issue --JimWae (talk) 23:26, 10 August 2009 (UTC)

"Whether "Euclidean" mathematics is an accurate model of "reality" (whatever that means) is a separate issue" Nonsense. Zeno's Paradoxes are about the movement of physical things. 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. Simple.Steaphen (talk) 23:33, 10 August 2009 (UTC)

Whoa there Jim! You have not established that we can "precisely and accurately" calculate the place and time. I'd enjoy seeing the mathematics that did that, in actual physical practice, not just theory. You realise of course that in doing so you'll have broken the Uncertainty Principle of quantum theory. Once again, if you have the physics to show how quantum physics (a real science, not just speculation and supposition) can do this, I'm very interested to learn.

Recorrecting. Mediation next step!Steaphen (talk) 23:28, 10 August 2009 (UTC)

We need not presume that time & space are the types of ontological entities that can be divided indefinitely (like the real number system can) to be able to calculate AN of answer using ordinary math ( w/o calculus). I contend it is a mistake to think that time & space can be "divided" at all - except as a conceptual exercise - they are not ontological entities like material objects are. Hold your own horses. Ordinary mathematics does not care about quantum theory; it is a system unto itself (whether it is a perfect model of "reality" is another matter). The point is that the calculation CAN be done (yes, even precisely) - and that still the issue is not resolved. That is what make it a paradox. If we do not point out that the calculation CAN be done, then the paradox is much less paradoxical. If you think what you have put in there is a *correction*, you are sadly mistaken - as you will see if you insist on mediation to determine how opaque your "correction" is. --JimWae (talk) 23:46, 10 August 2009 (UTC)

Using ordinary mathematics we CAN arrive at a precise calculation regarding the space & time when/where they meet. Whether that answer "mirrors" reality is what makes the paradox knotty. --JimWae (talk) 23:50, 10 August 2009 (UTC)

It doesn't make it "knotty", just wrong. Plain and simple. As Feynman said, "the simple ideas of geometry extended down into infinitely small space is wrong." Wrong content has no place in an encyclopedia.Steaphen (talk) 00:04, 11 August 2009 (UTC)

A calculation does not imply a "solution"--JimWae (talk) 23:51, 10 August 2009 (UTC)

Calculations using horoscopes and other superstitions can also be used. I don't see any credible reason for your bias to use calculus. (apologies) "algebra" !

Let's get some independent adjudication on this.  :) 124.189.34.79 (talk) 01:34, 11 August 2009 (UTC)

Please try to read my comments more carefully. It is the paradox that is knotty, not Euclidean geometry. I have not staked any of what I have said on calculus at all - I have repeatedly pointed out that calculus is NOT needed to get a precise (tho not necessarily accurate) answer -- just algebra --JimWae (talk) 00:23, 11 August 2009 (UTC)

(Reply by Steaphen):

And your point is? (pun intendend).

Irrespective of whether you argue for algebra, the fundamental issue still stands .. .the inapplicability of using any mathematics that is reliant on continuity to solve Zeno's Paradoxes. If such were the case we could dispense with quantum theory and simply use algebra to totally, accurately and precisely determine and predict the movement of quantum stuff (and ipso facto, the accurate and precise movement of hares, runners et al).

Jim, you appear to echo the disconnect of others with regards to the simple fact that physical movement has been found to be fundamentally discontinuous. If you have any evidence of its "endless" continuity, please enlighten the scientific community. That reminds me, where's my Nobel Prize! In the section Talk:Zeno's_paradoxes#Not_seeing_the_trees_for_the_forest I've disproved either the Heisenberg's Uncertainty Principle, or the validity of using infinite series to solve Zeno's Paradoxes. Which one do you wish to junk?

Let's see what the adjudicators say Steaphen (talk) 01:52, 11 August 2009 (UTC)

Do you really wish to maintain that it is an "error" to say that "using algebra, we can derive a (PRECISE) time & place at which Achilles catches the tortoise"?

Jim, your responses are getting embarrassing (for you). Do you not understand that according to quantum theory, there is nothing that can be precisely determined. Any macro-sized object will appear to be in a particular place, at a particular time. But drill down, and look at, say, an atom in the tip of the toe of Archille's front foot. Can you precisely know its location at a particular time, and by extension, that of all the atoms in his body?

This has been the glaring disconnect of basically all the respondents at this site - dsconnecting the proven science of quantum theory from some fanciful wish as to how they (and you) would like to think reality behaves.

So to answer your question: absolutely, categorically YES, it is WRONG to say that "using algebra, we can derive a precise time and place at which Archilles catches the tortoise." The main article needs to reflect this, for reasons outlined in the mediation request Misplaced Pages:Mediation Cabal/Cases/2009-08-09/. Steaphen (talk) 04:26, 11 August 2009 (UTC)

Neither space nor time are, in themselves, either continuous or discontinuous. Space & time (a la Kant) are unavoidable mental constructs with which we shape our world. We can both divide & separate matter. We can "divide" both space and time, but we can separate neither.--JimWae (talk) 03:30, 11 August 2009 (UTC)

Informal mediation requested Misplaced Pages:Mediation Cabal/Cases/2009-08-09/ Steaphen (talk) 03:32, 11 August 2009 (UTC)

Please stop breaking up my comments

Heisenberg Uncertainty Principle states that we cannot determine the position and momentum of microscopic particles of matter at the same time.

You are treating space and time as if they are analogous to matter. Your view is not supported by references to works in the field of quantum theory.

There is no denying that using algebra, a precise numerical answer can be calculated. It is another matter as to whether it gives a false precision. Before you can argue that it is false precision, it needs to be established that the calculation IS precise --JimWae (talk) 05:34, 11 August 2009 (UTC)

Btw, in case you have not noticed yet, I have never claimed the math solves the paradoxes. The section we are arguing about is entitled "Proposed solutions". There used to be a section for "issues with the proposed solutions" - but it lacked references. --JimWae (talk) 06:15, 11 August 2009 (UTC)

My apologies for the insertions. As I understand, Heisenberg's Uncertainty Principle applies to all conjugate quantities, including time-energy, and not just position-momentum. Once again, a precise value (location, time) cannot be meaningfully calculated, anymore than can the number of angels on pinheads. The Uncertainty Principle applies to large bodies as well. Even though some use simple ideas of combined mass to work out that the wavelength of the object becomes so small as to be unnoticable in everyday life, it does not matter how unnoticable or short the wave-length. It is not infinitely short!

Jim, to clarify - the infinite-series solutions ignore or deny the fundamental wave-particle duality of all matter (and all conglomerations of matter). Any large object will exhibit wave-like behaviour (according to the Wikipeida article wave-particle duality, this has been done for C₆₀F₄₈, a fluorinated buckyball with a mass of about 1600 u, composed of 108 atoms) which means the precise position and time of any object, even planets, cannot be precisely calculated or determined (even in theory).

Despite the fact that the calculated wavelength for a runner, tortoise or arrow is unnoticeably small (short), it is still some finite wave-length. Thus, infinite-series or simple algebra is most definitely not able to be used to calculate precisely where and when Achilles will overtake the tortoise (even in theory).

The widely-accepted infinite-series solutions to Zeno's Paradoxes are clearly incorrect, lacking compatibility and congruency with the experimental facts, and the main article needs to reflect this.Steaphen (talk) 00:40, 12 August 2009 (UTC)

There is of course one one exception to what I have stated above: you can apply infinite-series solutions (simple algebra) to precisely calculate the position and time of an object ... the only problem is that the mass of the object must be infinite (thus having an infinitely short wave-length, enabling precise location). Actually, there's another problem, the infinite-mass runner would need infinite energy to start his race, as would the bowman who attempted to pick up and shoot an infinite-mass arrow. :) Steaphen (talk) 01:05, 12 August 2009 (UTC)

As for your reference to space-time, the mediation request does not mention space-time, merely that infinite-series should not be mentioned as having solved the paradoxes. The main reason for the request was your re-editing of my change of sentence from "we can calculate" .. to "we can approximate" and others that implied, inferred or stated infinite-series solved the paradoxes.

The thought-experiment provided in the section Talk:Zeno's_paradoxes#Not_seeing_the_trees_for_the_forest requires you either accept quantum theory, and dismiss infinite-series solutions or vice versa. If you have knowledge of how and why the Uncertainty Principle does not apply, please enlighten all of us. In which case, the Nobel is yours. It's a done deal.

But I remain curious. Just what is your opinion of those physicists who hold such esoteric, non-mechanistic views such as physicist R.C.Henry (see above)? Or Yale's Emeritus Professor of Physics Margenau (who explained that "...each of us is the Universal Mind but inflicted with limitations that obscure all but a tiny fraction of its aspects and properties.") or Schrödinger who wrote: "There is obviously only one alternative, namely the unification of minds or consciousness. Their multiplicity is only apparent, in truth there is only one mind".?

Do you believe they would (or do, for those still living) claim or agree that "simple ideas of geometry (can be) extended down into infinitely small space"? Perhaps you should ask them.

Steaphen (talk) 09:47, 11 August 2009 (UTC)

---

Two Trains and a Fly: Two trains are traveling towards each other on the same track. One train is traveling at 30 km/h, the other at 20 km/h.

When the trains are 100 km apart, a superfly leaves one and flies towards the other. When it reaches the other it immediately travels back to the first. Upon returning, it immediately starts back to the second, and so on, until the trains crash.

Question: When the trains crash, how far has the superfly travelled, if it flies continuously at 50 km/h? Or, if you prefer, what is the upper-bound?

Mediator on deck!

Hi folks. I'm willing to help out as a mediator here if you're still interested. I'm not entirely across the issues, but if I understand correctly so far, there is essentially a disagreement about a clash of Zeno's paradox and QM... is that right? Could someone give me a brief summary, from a content perspective, of how this impacts on the article? i.e. are there disputed sources, undue weight issues etc. Cheers, Blippy (talk) 05:43, 16 August 2009 (UTC)

Blippy, why have you responded here, and not on the mediation page Misplaced Pages:Mediation_Cabal/Cases/2009-08-09/?

In answer to your question, the issue is one of undue weight, unsupported assumptions, and clear errors (based on simple analysis (see Proof 1)).

See the discussion section on Misplaced Pages:Mediation_Cabal/Cases/2009-08-09/ for details

Steaphen (talk) 08:14, 16 August 2009 (UTC)

Hi Steaphen. This is my first mediation - I'm used to 3O's - when I looked I saw that some do it this way, others on the mediation site. I'm happy to move the conversation there if you prefer. Can I slow things down a smidge and deal with one thing at a time? I note that that your first point concerns whether the ZP remains a problem. I appreciate that you are providing reasons why it is still problematic, but do you have a WP:RS for that assertion as well? I just want to make sure we tick all the boxes - so to speak. Cheers, Blippy (talk) 08:19, 16 August 2009 (UTC)

Blippy, I appreciate your interest in ticking the boxes, but the problem is this article (Zeno's Paradoxes) is in a class of its own for some very important reasons.

Let's consider "ticking the box" regarding "Reliable Sources" - in any era, the prevailing cultural belief-system will form a filter for acceptable ideas and behaviours. For example, the official verdicts of the Salem witch trials that saw 18 people executed for having practised "witchcraft", were "peer (court) reviewed" judgements (and that's official!).

Then there was the official "peer (church) reviewed" dogmas that saw Giordano Bruno burnt alive, and Galileo spend the remainder of his later years under house arrest.

Placing our faith in the Gospels of science (The Book of Mathematics) is really no different to putting our faith in the Gospels of religion. The superstition that movement is "perfectly continuous" and that infinite-series solve the paradoxes has parallels to the witch-burning attitudes of past eras.

What would navigate us out of the ruff, so to speak, is asking questions of the evidence, and to seek answers not based on the majority (crowd) opinion, but on good old-fashioned wisdom - timeless principles that will stand the test of time. What is a timeless principle? Ask the question, see what answers you get.

So in answer to your question, yes I have Reliable Sources (some of which have been mentioned on this talk pages), but I think, given the witch-hanging/burning penchant of people through the ages, I'm inclined to suggest that we play down the importance of crowd opinion.

The reason this article (Zeno's Paradoxes) is in a class of its own, is that it goes to the core of the fundamental nature of our physical reality, upon which various assumptions (and crowd opinions) are then formed. Get the base understanding wrong (i.e the nature of physical movement, and space-time), and all the following assumptions and theories become based on a house of cards, held in place only by dogmas and opinions held by scientists, philosophers and mathematicians.

Steaphen (talk) 10:26, 16 August 2009 (UTC)

Hmmmm, that's going to make it tricky then Steaphen. I'm all for wisdom and not being swayed by public opinion, or even scientific opinion come to that. But what you're wanting to do is to change a WP article in a particular way. Whilst I acknowledge that ZP may have fundamental, perhaps axiomatic and even ontological implications, perhaps WP is not the place for such wisdom to be imparted? The problem is that WP has a very distinct hang up on reliable sources and not promulgating original research. This is kind of handy though, because it stops all those Zeno deniers from running amok too :-) So I think that given we're discussing this in the context of WP, we'll need to play by the WP rules - which may preclude your insights from being utilised here. How does that sound? Cheers, Blippy (talk) 10:43, 16 August 2009 (UTC)

Blippy, my point is not to ignore Reliable Sources, but in how we select what is reliable, as the selection will be filtered based on one's world view. Physicist David Bohm expressed quite clearly that "according to the quantum theory, movement is not fundamentally continuous." In that case, infinite-series solutions, based on the assumption that physical movement IS continuous are invalid. But those who wish to maintain "movement is continuous" views will find reasons to dismiss David Bohm as a reliable source. Do you get my point?

Another way to consider this, is that the root assumptions we hold about physical movement will colour our perceptions. So, for the sake of "ticking boxes" perhaps a way to approach this is to be quite clear about those root assumptions, and the beliefs and theories that result from those root assumptions. At the very least, the fact that the root assumption of continuity has been taken for granted very much needs to be acknowledged.

Another dimension to this issue is your response here. Do you believe physical movement is "perfectly" continuous? Movement either is "perfectly continuous", or it isn't. It's a yes-no consideration, but how you answer will shape how you deal with this issue, and whether as a moderator you shift the issue towards "maybe we have to preclude your insights" etc... they're not originally mine, so they're not my insights. Are you suggesting that Bohm is not a reliable source, because your world view is one of continuity of movement? In other words, as a moderator, are you seeking to inject your bias in this matter?

I sense that your comment "because it stops all those Zeno deniers" (whatever that means), indicates your bias towards a "movement is continuous" world view, and therefore perhaps precludes your dealing with this issue. Steaphen (talk) 12:47, 16 August 2009 (UTC)

Hi Steaphen. I'm actually quite taken with Bohm's implicate order. But I'm not here to offer an opinion, rather mediation. And we can't get the mediation underway if we don't have at least two possible alternatives to choose from. We need to have RS's backing each of the contested positions, otherwise there is no contest - WP is only here for RS material. So let's start there - I don't recall Bohm dealing with Zeno's Paradox explicitly. Can you direct me to the source for that? Cheers, Blippy (talk) 13:34, 16 August 2009 (UTC)

"Reliable Sources" are many, but again, I'm keen to see how you select which are reliable and which are not. For example physicist Fred Alan Wolf explains in his award-winning book "Taking the quantum leap" how movement is discontinuous and that Zeno was right ("Werner Heisenberg was ... awarded the Nobel prize in physics for his realization that Zeno was correct after all"). Norman Friedman, author of "The Hidden Domain" explains how reality is flickering on and off, and so on. There's plenty of authors/physicists who concur with the basic understanding that movement is not fundamentally continuous. In fact, I'd enjoy seeing any physicist stating categorically that he/she believed that movement WAS entirely continuous. I'd like to see that.

Do you have any reliable sources claiming that physical movement IS entirely continuous, which would be necessary if one is to apply infinite-series to physical movement?

Let's do a quick whip-around, "Hands up all those physicists who believe physical movement is perfectly continuous" ... and please provide details with which university or research institution you're employed, and how long you predict you'll remain employed.Steaphen (talk) 16:32, 16 August 2009 (UTC)

Blippy, since you asked the question, what reliable sources do you have that supports the idea that physical movement is entirely continuous, including continuity through sub-Planck scaled increments (e.g. of Zeno's arrow)? If you can't answer the question that you've asked, perhaps you should disqualify yourself from further mediation and/or that I bump this to a request formal mediation Steaphen (talk) 16:48, 16 August 2009 (UTC)

Excerpt, from "Taking the Quantum Leap", (Publishder, Harper and Row, New York), winner of the American National Book Award, 1982, by Dr Fred Alan Wolf (regarding the assumed continuous movement of Zeno's arrow):

"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."

Blippy, if you can't cite a reliable source (within say one week from today's timestamp) that refutes Wolf's view, or offers a credible alternative to the idea that physical "movement is not fundamentally continuous", I'll bump this to formal mediation, since informal mediation has failed to resolve the issues I've raised in Misplaced Pages:Mediation_Cabal/Cases/2009-08-09/.Steaphen (talk) 23:15, 16 August 2009 (UTC)

• Here and here are the edits that brought on the request for mediation. Algebra does not give an *approximate* answer. The section is proposed solutions, and the proposed solution should be allowed to be presented on its own terms. Since that edit, I have changed the wording to "Using ordinary mathematics we can arrive at a specific time when and place where Achilles would be able to catch up to the tortoise" towards eliminating the suggestion that the answer ordinary math & algebra gives is necessarily THE answer. There is no denying that algebra gives us AN answer that is as precise as the measurements of distance, speed and time are. Admittedly the answer is a false, overly-precise answer if the measurements of speed and distance and time are themselves approximations. It does, however, point out that neither calculus nor the sum of an infinite sequence is even needed. The limts of the algebraic solution is something that can be developed within the article later on -- without interjecting quantum theory even before the proposed algebraic solution is presented. The algebraic solution also applies ONLY to Achilles and the tortoise - and not to any other of ZPs. --JimWae (talk) 00:47, 17 August 2009 (UTC)

Your comment "Using ordinary mathematics we can arrive at a specific time when and place ...etc" is simply rehashing old ground. As I've named you as one of the parties involved (in the mediation docs), I'll extend to you the same courtesy of allowing time for you to cite reliable sources to support your assumptions, before requesting formal mediation.

Blippy, Jim, if a week is insufficient for either of you to cite reliable sources, what time-frame would you require? If you don't believe you can cite any reliable sources (asserting that physical movement is entirely continuous), let's not waste further time or dialogue and bump this straight to formal mediation, followed by, if necessary, arbitration.Steaphen (talk) 01:38, 17 August 2009 (UTC)

• The point is that the article should not take a position on whether movement in general is continuous or not - nobody knows, despite what a few may assert. The TITLE of the SECTION we are dealing with is "PROPOSED SOLUTIONS"--JimWae (talk) 04:38, 17 August 2009 (UTC)

Jim, you have made certain statements as fact without basis - they are your personal opinions (and thus POV). It is irrelevant which section such assumptions-as-fact are made, reliable sources are rquired to substantiate such claims. If you can't supply any reliable sources to back up your "facts" (e.g. that we can calculate exactly where Achilles overtakes the tortoise etc) then let's not waste further time. I'll request formal mediation unless you can supply reliable sources to backup your statements. Keeping in mind that these reliable sources will need to assert that movement of physical things (e.g. Zeno's arrow) is perfectly continuous, as opposed to Wolf's asserting that it is not (perfectly continuous)124.189.34.79 (talk) 04:52, 17 August 2009 (UTC)

• Simple mathematical calculations that anyone can follow do not require sources. Furthermore, this proposed simple mathematical solution can be found in nearly any discussion of ZPs. At the most, all I need to do is to find one reliable source that presents the same proposed argument. Algebra does not give us an approximation - though the limiataions of measurement do. However, the measurements are not given as approximations in the example - and that too is worth further discussion--JimWae (talk) 05:02, 17 August 2009 (UTC)
• Furthermore, the argument that the algebraic solution does not solve every issue with Achilles and the tortoise does not need to involve quantum physics - much simpler, more accessible argumaents are available --JimWae (talk) 05:10, 17 August 2009 (UTC)

I'm not interested in what the crowd says about your assumptions. What reliable sources can you cite that says "movement is fundamentally continuous" in order for any mathematics (based on continuity) to be relevant and applicable to understanding and calculating the when and where of said movement?

Jim, with respect, I'll not respond to any more of your opinions. As a courtesy, I'll allow one week, unless you request more time to find a reliable source that confirms "movement is fundamentally continuous, which thereby enables and validates the use of mathematics that are based on algebraic continuity".Steaphen (talk) 05:20, 17 August 2009 (UTC)

• Please see this article (which is listed as an external link in the article) - especially the 4th point. That is all that is needed to show the argument is RELEVANT --JimWae (talk) 05:28, 17 August 2009 (UTC)

Your referenced article says "Our belief that the mathematical theory of infinity describes space and time is justified to the extent that the laws of physics assume that it does," and you believe this qualifies as a reliable source? That if others assume something is true, and admit their assumptions, then that's good enough to assume it is true, and fact? That attitude and logic lands us all in serious witch-hanging, heretic-burning territory. See you in a week. CiaoSteaphen (talk) 05:50, 17 August 2009 (UTC)

• All I have to do is provide a source that demonstrates the algebraic argument is relevant. I do not need to show that the argument is the correct & complete solution. If you could realize this, we could get on with improving the article--JimWae (talk) 05:59, 17 August 2009 (UTC)
• It is not our task to present "the true nature of reality" to the reader. The best we can do, without doing WP:OR is present what reliable, and preferably scholarly, writers have to say on the subject--JimWae (talk) 06:09, 17 August 2009 (UTC)

Whoa!!

Hi JimWae and Steaphen (also 124.189.34.79?). Could I suggest we back up a bit here? Steaphen, you seem to think I'm here to refute or contribute to the content in some way - that's not my intention. You folk all seem eminently far more expert than I to take care of the content. It looks like you have a number of reliable sources Steaphen, would you agree JimWae? If so, is there any objection to changing the initial wording that Steaphen has raised that implies that there is no longer any controversy about ZP's? Cheers, Blippy (talk) 09:57, 17 August 2009 (UTC)

That was never in dispute, and the article has long stated that some hold there's still controversy. I do not think it ever said that scholars agree the controversy has been resolved - quite the opposite --JimWae (talk) 18:15, 17 August 2009 (UTC)

OK, thanks JimWae. So Steaphen, are you happy to leave the first section as is, or do you have some specific changes to the wording you would like to see? Cheers, Blippy (talk) 09:24, 18 August 2009 (UTC)

However, isn't Steaphan taking the position that (according to him) quantum physics has solved ZPs and that (some) quantum physicists hold the matter is closed? There is an important difference between establishing that the smallest theoretically-possible measurement of space is one Planck length and establishing that the smallest possible unit of space is one Planck length.

The original impetus for the mediation was the statement "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 incorrect. With mathematics we can only approximate place and time. The other issues raised on the Mediation page have not been addressed either. Given Jim's (and others') expected continued obstinacy in the face of the facts, and your (Blippy) apparent misunderstanding of the issues, I'm happy to move this matter forward to formal mediation.

This issue is not a trivial matter. A shift in belief-systems is inevitable (and arguably well overdue). Limiting, reductionist scientific beliefs (and their related religious and political beliefs) are causing increasingly adverse consequences for all of us. Yes, that's correct, reductionist scientific world-views give rise to fundamentalist religious beliefs - they are sister-belief systems, both objectify the causal agent for self-organisation of "systems" (physical and spiritual, resp.). That disconnect (objectification) is the cause of most if not all of the world's ills. And that disconnect is maintained by adherence to beliefs as exemplified on the main page of this article.Steaphen (talk) 08:35, 20 August 2009 (UTC)

For the sake of clarity, and without recopying the text from the mediation page, any statement that implies mathematics can give correct (precise) location and speed of arrows, runners,tortoises is, based on the evidence, incorrect. Arguments to the contrary are simply highly biased points of view, speculative, and have no demonstrable basis in this reality. If you can cite reliable sources that state mathematics (in any form) can give precise physical details of objects, (and in the process disproving the Uncertainty Principle) then I'd be highly interested to see them. Since none will be forthcoming, I'll expect the change in the text as outlined on the mediation page.Steaphen (talk) 08:48, 20 August 2009 (UTC)

I'd like to make an observation at this point... mediation (formal or informal) strives to reach a point where all parties are happy with the outcome. You have raised a number of issues Steaphen, and as an independent third party, I am attempting to address them in a particular sequence. It is simply not possible to address all of the issues simultaneously. If you feel that I am misunderstanding something I am more than open to being told what that is, however large swathes of texts repeating contentious points will not help. I would like to proceed in an orderly fashion, identifying areas where we can find immediate agreement, and "creep up" on the bigger issues. If that doesn't suit I'm happy to bow out. Cheers, Blippy (talk) 10:43, 20 August 2009 (UTC)

I would question your status as an independent third party, given your earlier reference to "Zeno deniers", "leaving the first section as is" (when I had outlined the errors contained therein), and calling for me to cite reliable sources while not requesting reliable sources from those who had objected to my edits. I note that you have yet to call for reliable sources from Jim. These factors together would seem to indicate bias, and/or inexperience with this process, and/or not understanding the issues, leading me to expect that formal mediation will be a necessary next step.

Blippy, assuming for the moment your good intent (sans bias/prejudice), let's clarify matters: I've provided a reliable source (winner 1982 National Book Award), explaining that we cannot precisely determine the position of an arrow. To counter this reliable source, I'd expect that you call for a reliable source (from Jim, et al) that does claim we can precisely determine the position of the arrow. And by "precisely" that means stating that even at quantum scales and sub-quantum scales (since the arrow will at some stage travel through such increments, since the arrow must traverse infinite points, according infinite-series solutions) we can precisely and wholly determine the arrow's position and momentum. Totally, without error, and most importantly, without uncertainty. Perfect reductionism, as required by infinite-series solutions.Steaphen (talk) 14:38, 20 August 2009 (UTC)

Blippy, as you suggested, one thing at a time, so lets start with the first one:-

I suggest the paragraph in the first section on the main page that reads:

"Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found most proposed solutions somewhat unsatisfactory. More modern methods using calculus have generally satisfied mathematicians and engineers. Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see Thomson's lamp) continue to produce at least temporary puzzlement in elucidating what, if anything, is wrong with the argument."

be changed to:

"Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found most proposed solutions somewhat unsatisfactory. More modern methods using calculus have generally satisfied mathematicians and engineers. However, in 1926 physicist Werner Heisenberg formulated the Uncertainty Principle which disallows precise calculation of the speed and location of physical objects such as arrows, runners and tortoises. Variations on the paradoxes such as Thomson's lamp, together with the Schrödinger's Cat Paradox and the EPR Paradox provide further puzzlement for physicists and philosophers."Steaphen (talk) 02:11, 21 August 2009 (UTC)

1> When has this proposed paragraph ever been discussed previously?? If not, why is a mediator needed? 2> Please elucidate how Schrödinger's Cat Paradox and the EPR Paradox are "variations on the ZPs". 3> Heisenberg is not needed to cast uncertainty on measurements. 4> re Heisenberg - the principle is about simultaneously determining location and momentum, and applies much less to macroscopic objects 5> Heisenberg is not about calculating, but about determining. You've mixed-up the concepts again 6> Precision does not have the same meaning as correct. "I am 33.345678934503 metres tall" is very precise and very incorrect --JimWae (talk) 04:32, 21 August 2009 (UTC)

It's been discussed before, dating back about 2 years, if you recall.

If you were to attempt to reflect on this matter, you would find all paradoxes relating to the movement and reality of physical things share the same base issues, Zeno's included. The question "is the cat alive or dead", is the same as "is the arrow or Achilles actually there or not"? It relates to the fundamentals of quantum theory (which presumably you've not bothered to acquaint yourself with). And I wrote, "together with" meaning they are not specifically Zeno's Paradox, but nonetheless such paradoxes remain unsolved, as do Zeno's.

We cannot determine (or calculate) precisely both speed and location of anything, no matter what its size. Your reply that "applies much less to macroscopic object" beggers belief.

"Precision" is precisely what the solutions to Zeno's Paradoxes are about. I could say any number of things and argue they are close enough, so its good enough. They're not. As far as mixing up concepts, interesting response. I can argue that "precisely calculating" the number of angels on a pinhead indicates when Achilles will overtake the tortoise, and in some universe or other, I might be close enough to the truth. Who knows, because you are most certainly not referring to facts in your responses.

By the way Blippy, where's those reliable sources stating that we can precisely calculate (including 100% accuracy at quantum and subquantum scales) the exact location and speed at which Achilles is running, or the arrow is flying? By your own words, "We need to have RS's backing each of the contested positions, otherwise there is no contest - WP is only here for RS material. So let's start there" - so again, what reliable sources have you gained from Jim stating that we can totally, precisely and accurately calculate the actual location and speed of an object as it passes through the quantum scaled increments (that they inevitably MUST do, according to the requirements of infinite-series solutions, or any solution that is based on the implicit assumption of continuity?)Steaphen (talk) 05:18, 21 August 2009 (UTC)

If you re going to repeatedly use precision and 'accuracy interchangeably, we will never get anywhere in this discussion. Please acquaint yourself with the differences --JimWae (talk) 05:13, 21 August 2009 (UTC)

From Accuracy and precision "In the fields of science, engineering, industry and statistics, accuracy is the degree of closeness of a measured or calculated quantity to its actual (true) value. Accuracy is closely related to precision, also called reproducibility or repeatability, the degree to which further measurements or calculations show the same or similar results"

Blippy, what reliable sources have you gained stating that we can precisely calculate the actual, true value of an object's momentum and position, and that we can reproduce those calculations to show the same results (of their true values)?Steaphen (talk) 05:28, 21 August 2009 (UTC)

OK. I'm going to stick pretty much to the incremental approach here folks and not get sidetracked by issues yet to be addressed. I'm not sure whether you are wanting me to stay involved or not Steaphen. I'll assume the former at this stage. So, I note that in your preferred version the first issue is no longer disputed. You are happy to leave the first two sentences as they are. I presume you don't have an issue with this JimWae? That gives us Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found most proposed solutions somewhat unsatisfactory. More modern methods using calculus have generally satisfied mathematicians and engineers. as agreed text. Yes? Cheers, Blippy (talk) 07:32, 21 August 2009 (UTC)

Agreed, but I wonder why we continue dialogue given your earlier request "We need to have RS's backing each of the contested positions, otherwise there is no contest - WP is only here for RS material." On that basis, there's no contest, unless Jim can provide a reliable source (a physicist) who's willing to commit career suicide and state that we can precisely and accurately calculate an arrow's or Achilles' momentum and location, at and below quantum scaled increments in the arrow's flight and during Achilles' run. There's plenty of reliable sources saying we can't. I'd like to see one that clearly says we can. Recognising of course that such a statement would necessarily (either implicitly or explicitly) refute the Uncertainty Principle, since such a statement would also affirm that the motion (speed and location) of any atom or particle within the arrow or Achilles could also be precisely and accurately calculated.Steaphen (talk) 23:52, 21 August 2009 (UTC)

We're getting there Steaphen, just slowly, slowly. Now, the first two sentences are fine, but you dispute the rest of that paragraph. So, firstly, what is it about this bit Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see Thomson's lamp) continue to produce at least temporary puzzlement in elucidating what, if anything, is wrong with the argument." that you dispute? Is there a lack of RS to back this up in your view, or something else? I know you wish to replace it with your preferred text, but why do you wish to delete this, specifically? Cheers, Blippy (talk) 02:21, 22 August 2009 (UTC)

The sentence "Zeno's paradoxes were a major problem for ancient and medieval philosophers" implies they aren't now. Turn the sentence around, if you like, or include the "however".. either way, the bias has not yet been addressed.

For example, we could say "Zeno's paradoxes have remained a major problem for philosophers since their origin. Many mathematicians using calculus hesitate to say that all paradoxes remain unsolved." What "reliable sources" have been cited to argue that they have been solved? And in solving them, how specifically did they address the uncertainty principle issue raised previously? Blippy, do you understand the bias implicit in the first sentence? If the first two sentences stay, then you will need the "however, in 1926 ..." or similar to address the biasSteaphen (talk) 07:36, 22 August 2009 (UTC)

Sadly Steaphen you are now wanting to change what you had just said you agreed to!! OK, so the very first sentence is back in dispute. Do you disagree with the substance of the sentence i.e. that ancient and medieval philosophers struggled with ZP's, or just the implication that they are no longer a problem, or both? Cheers, Blippy (talk) 07:53, 22 August 2009 (UTC)

The point of the reply is that if the first sentence stays (to which I agreed, as a first step) then you would need the "However, in 1926 ..." or similar to address the bias. I see no reason to be sad, other than for you to do your job, and address the bias. Perhaps though I do need to take you to task, because you have not directly addressed the first issue raised in this Misplaced Pages:Mediation_Cabal/Cases/2009-08-09/ mediation dispute. Allowing your first request (and my initial "agreed") was a small gesture of good faith, in that you would at some point seek to address the bias implicit in the disputed sentence. Perhaps you took my "agreed" reply as a fait accompli, independent of other sentences or issues. If so that was your incorrect assumption.Steaphen (talk) 09:15, 22 August 2009 (UTC)

• The sentence "Zeno's paradoxes were a major problem for ancient and medieval philosophers" DOES NOT IMPLY that they are not now, though some with an inclination to jump to conclusions might infer so. It merely asserts what is agreed upon. For wikipedia to say they they REMAIN a problem would be a problem, since people disagree on that. Instead, the article does not take a stand on whether they actually remain a problem today & presents the differing viewpoints re the present on whether they are still a problem.
• The other paradoxes are similar to ZPS only in being paradoxes. If the only standard for being similar is that they be paradoxes, we could include the Twin Paradox too--JimWae (talk) 19:24, 22 August 2009 (UTC)
• "However, in 1926..." actually has wikipedia appear to take a position that in 1926 something happened to change whether the ZPs were still considered a problem -- but it remains glaringly ambiguos on whether the effect of the uncertainty principle was on the ZPs or on the calculus approach. The "however" is uncalled for. The relevance of the Uncertainty Principle is left dangling--JimWae (talk) 19:26, 22 August 2009 (UTC)

• I have made a change to the lede to give more prominence to developments in physics regarding uncertainty --JimWae (talk) 20:56, 22 August 2009 (UTC)

Blippy, I suggest, "Zeno's Paradoxes have been a problem for philosophers since Zeno's time" (to replace the entire paragraph in dispute). If there are any reliable sources (physicists) who now state that they aren't, perhaps you could call for them to be presented here. I have presented one who says they are. Any by "reliable" I mean a physicist, not some mathematician off in cloud-cookoo land who has lost connection with physical reality, and with any congruent, verifiable theories thereof.Steaphen (talk) 21:37, 22 August 2009 (UTC)

Jim, A reliable source says "it was natural to imagine continuity as 'made up' of an infinite number of still frames, even 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.

What part of "demolished" do you have problem with? To have Misplaced Pages include material that suggests that such "imaginings" have not been demolished, requires a reliable source who says they have been verified in fact. Simply citing 'noise of the crowd' is a disservice to yourself and Misplaced Pages. The case 'for' (that the Paradoxes remain unresolved) has been made. Now the case 'against' needs to be made, including reference to reliable sources.Steaphen (talk) 23:03, 22 August 2009 (UTC)

Blippy, the unbiased observer (of this discussion) will note that while I suggest amendments on this discussion page, Jim proceeds to make edits to the main article based on his beliefs, as if to suggest he is an authority on such matters. And all the while, doing so without referencing reliable sources. If you can't rein in the bias, and the clear breaches of Wikipeida policy, what point this mediation?Steaphen (talk) 22:14, 22 August 2009 (UTC)

Blippy, I've given this mediation process more time than is perhaps wise, allowing it to affect my focus on more important and relevant business matters. I won't spend more time responding to the style of responses so far. You'll need to cut to the chase, request reliable sources that support Jim's arguments, and definitively sort out the bias that I've highlighted. A lack of reliable sources (stating, as previously explained, that experimental and theoretical physicists can accurately and precisely, in theory and practice, determine the exact motion of physical objects, including arrows and the like) will confirm this mediation has failed.Steaphen (talk) 01:18, 23 August 2009 (UTC)

Explanation for my recent edits

My recent edits were reverted by JimWae (talk · contribs). 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—more importantly—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 {{fact}} 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 Misplaced Pages 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 1-1/2 and stops at 1-1/2. 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)

• Could you locate for me more clearly the part you want to revise?--JimWae (talk) 06:58, 27

October 2009 (UTC)

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 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 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 Misplaced Pages: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 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)

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 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. 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.)

• 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.

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)

• 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: Misplaced Pages: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 Misplaced Pages:Talk page guidelines, the first sentence of which reads:

The purpose of a Misplaced Pages 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 ☎ 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 ☎ 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:

1. "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)
2. "that space is continuous is, I believe, wrong." Professor Richard Feynman, The Messenger Series: Seeking New Laws
3. (according to some estimates) "our universe is flickering on and off every 5.3 x 10 seconds", Norman Friedman, The Hidden Domain: Home of the Quantum Wave Function, The Woodbridge Group, Eugene OR 1997, page 165.
4. "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
5. "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
6. "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).

7. "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.

• 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:

Steaphen (talk) 23:29, 24 November 2009 (UTC)

Comments and opinions on the above section

This an interesting development. 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

• 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)

Hope it helps. Ansgarf (talk) 13:37, 24 November 2009 (UTC)

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 Misplaced Pages board of your comments here, suggesting you've opened Misplaced Pages 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 Misplaced Pages.

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 Misplaced Pages 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 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.

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)

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 :${\displaystyle \lambda ={h}/{p}}$ (where p = momentum). The infinitesimal precision of the object's position or any part thereof (as required by infinite-series solutions) requires that ${\displaystyle \lambda }$ approaches zero (since the de Broglie wavelength of the object indicates the range of possible positions and momentums of the object).

BUT as ${\displaystyle \lambda }$ 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 Misplaced Pages 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 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 ${\displaystyle \lim _{n\rightarrow \infty }\sum _{k=0}^{n}1/2^{k}}$ 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)

Resolution of bias and POV

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"The uncertainty in finding the point at which Achilles catches the tortoise comes more than 2 dozen magnitudes of distance before QM enters the picture. QM is an issue at around 10 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 seconds. We do not have instruments than get anywhere near the theoretical limits of QM, and introducing QM as the primary uncertainty in a race is absurd."

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

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

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

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

1. J. Madeleine Nash, "Unfinished Symphony", Time, December 31, 1999, page 61

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