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  • Post #24

Very interesting article Mr Harkopos.

I was under the assumption that Zeno's paradoix was implicitly resolved with the advent of differential calculus and the idea of limits.

i.e. the sum of an infinite series may converge to a finite value, which in turn represents the limit of the series. The infinitesimal sum of dx(1/dx/dt) approaches a finite value over any abritrary interval on R, prodived dx/dt is continuos, smooth, and exists on the interval.

The erroneous assumption implicit in Zeno's paradox is that the sum of an infinite sries is always infinite.

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    • Post #25

    Also, all of the variations of Zeno's Paradox also lack any self-consistency and self-referential integrity.

    For example, the argument put forth that in order to travel an interval AD one must first travel half that interval AB, and so on, leads to obvious problems.

    The lack of consistency arises when one states that once one reaches the first half-interval AB, another half-interval awaits and this progression continues indefinitely. One must therefore complete an infinite amount of actions and can never arrive at the end of the interval. The obvious problem here is, you managed to travel the first half-interval AB in a finite amount of actions without encountering infinity.That interval AB itself is arbitrary and contains an infinite number of half-intervals. Since the original intent is to show one can never reach the original interval AD in a finite umber of steps and actions, how is it you came to traverse the sub-interval AB? Based on the assumptions, an infinite number of actions would also have been required for that interval.

    This should immediately tell one that the something is amiss with the idea of actions on intervals as they relate to the infinite.

      Hi Bubba,

      Thanks for an interesting comment. There is no erroneous assumption in Zeno's paradox. If you get the reference I mentioned (Barnes) you will see that the fundamental premise of Zeno is that "nothing can perform infinite many tasks". Calculus does not resolve this issue. Calculus provides asymptotic convergence conditions for mathematics only. The limit of the series as you say, it is only reached asymptotically. In mathematics we call that convergence. But what convergence means in the case of physical motion in infinitely divisible space? I don't know. If you know, I would be interested to find out, namely, when does the body in motion exactly reach its end point.

      Barnes explains the whole issue well and also the argument of Aristotle which was the first and last viable argument against Zeno. Zeno's argument can be transformed to:

      1.Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps

      2.Supertasks are impossible

      3.Therefore motion is impossible

      There is huge literature on this subject and the concept of infinite supertask machines.

      Actually, Zeno's task can be modeled by a Grandi series, which converges to infinity:

      http://en.wikipedia.org/wiki/Thomson's_lamp

      http://en.wikipedia.org/wiki/Supertask#Zeno

      In order to resolve the paradox many assume that motion is possible and then declare the argument of Zeno unsound using modus tollens.

      However, it is the very possibility of motion that Zeno challenged.

      Note that solution that claim that as soon as motion starts it concluded because each subsequent motion takes less time, so that the time intervals converge to a finite value according to calculus, are naive because according to Zeno, motion cannot even start. Zeno's philosophy was that there is no such thing called motion. Everything is immovable, at rest, and what we see is an illusion. I think what we see may be virtual reality. In a virtual reality, motion is possible because it is pixelized. But again, what I think is not important. It is what experiments will show that is important. Talk is cheap in physics, almost.

      Thanks and regards.

      Hello Bubba,

      I apologize for an error in my post. I wanted to say that the Grandi series converges to 1/2 at infinity. However, at the same time, depending on solution, the series has no sum. Thus, the series diverges.

      So, instead of using the typical calculus approach which sets infinitesimal values to zero, the Grandi series approach should convince you that this is not a trivial problem.

      To illustrate this further, consider S = at^2/2, the known equation. Now, consider that the body moves ds in time dt. The equation becomes:

      s+ds = a(t+dt)^2/2 which with some algebra becomes:

      2s+2ds = at^2 +2atdt+a(dt)^2

      But as^2 = 2s, so we get: 2s+ds = 2s+2atdt+a(dt)^2, which reduces to:

      2ds = 2atdt +a(dt)^2 or ds/dt = at +dt/2

      But calculus tells us that ds/dt = at. How can that be?

      The mathematician response is that in the limit dt goes to zero. But is it exactly zero? If it is exactly zero, then S+ds will be forever equal to s and motion cannot take place. If it is not zero, then s = at+dt/2 and these sums of dt/2 diverge. Of course, mathematicians can always add a few more axioms and get anything they want.

      By this response, triggered by your well put argument, I want to show that calculus wants its cake and eat it too when it comes to justifying motion (although it describes the phenomenon correctly). Things are not that simple. There is a lot of work to be done to transform our naive views of reality to something more advance that can help us to progress.

      All the best.

      Efthimios

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        • Post #29

        As Einstein recognized, the calculus of continuous functions requires specifying boundary conditions. Because "From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter ..." Einstein's finite but unbounded, quasi-Euclidean model of relative matter rest states came mathematically complete, with the origin of inertia assumed at the boundary of a singularity, and otherwise unexplained. (The foregoing is picked up from the technical note in my essay "Can we see reality from here?")

        If motion is primary, as Mach believed, the origin of inertia needs no explanation. We know that this cannot be true, however, because also as Einstein recognized, because of the problem describing continuous function physics without singularities. So cosmology, once subbed as mere philosophy, has taken--if not the leading, at least an important supporting--role in physics.

        Tom

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        • Post #30

        Hi Efthimios,

        .

        btw, I found your article to offer a lot of food for thought and hope you score high on the final ballot. Best of luck.

        Anyway, regarding this issue, I don't think the problem here is one of infinity, it is the way we think about infinity.

        First of all, I think we need to keep in mind that mathematics simply represents a model for reality. You stated, "I want to show that calculus wants its cake and eat it too when it comes to justifying motion."

        I think it was Confucius who said, "Do not confuse the finger pointing to the moon for the moon itself." If we were to rely exclusively on mathematics to form our picture of reality, we would immediately find ourselves in a lot of trouble. This is why we check to make sure a mathematical solution to a problem is physically admissible. If the solution does not conform to what we observe then we do not throw out our observations, we reconsider the validity of the solution or we reevaluate our line of reasoning that led to the solution. Sometimes, a theorist cannot decide whether or not a solution to a problem is a physically admissible one(e.g. String Theory), but that is another story. The danger is relying too heavily on theory to form a complete picture of reality.

        Basically then, in the context of this discussion, motion is possible and exists because we observe it to exist. Any argument or solution that infers motion is impossible must therefore either be reframed or thrown out completely. There is no way around this. Observation and experiment always has the final say in science. So, we are on shaky ground when we expect our mathematics to justify motion.

        I think that the conceptual difficulties inherent in these types of arguments all lie in our ideas about casuality.

        IMO, the fundamental question is not why do things happen the way they do, but why does anything happen at all? This is where Newton, motion, and the concept of inertia comes in. If you think about it critically, the first law is logically necessary in any universe where cause and effect has meaning. This is because the first law essentially reduces to a simple statement about causation--nothing happens without an impetus to action or a reason behind the impetus. If this were not the case, we would simply have random chaos and unpredictability.

        The same applies to any idea of uniform motion--i.e. inertial. In the context of this discussion, there is not an infinite number of actions taking place in inertial motion because there is no action required. Uniform motion is relative and there is no preferred frame of reference. In one frame, an object may appear to undergo inertial motion. In another frame of reference, the object may appear to be at rest. If action was required to maintain inertial motion then motion would not be relative and there must be a preferred frame of reference.

        Zeno's Paradox,in all it's incarnations,therefore leaves out this concept of relative motion. In one frame of reference where an object is undergoing uniform motion and must travel the segment AB, one can find another frame of reference where the object is at rest and the length of the segment is 0. Which one corresponds to reality? Zenos paradox becomes a non-sequitur when relative motion is considered.

        Also, when you inquire-- ."But calculus tells us that ds/dt = at. How can that be?", the answer is because the term 'a' implies an impetus that acts to change unfiform motion or a state of rest relative to an observer. Implicit in the calculation is the concept of non-uniform motion. For each element dt, the object 'caries' with it the velocity it had at t-dt. Through impetus, an additional velocity was added to the object during the time element dt.

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        • Post #31

        sorry, forgot to attach my handle to last post. This is Bubba.

        Thanks for your detailed response. I agree with you about calculus but I disagree - for whatever it worth - with your statement that "Basically then, in the context of this discussion, motion is possible and exists because we observe it to exist." This is not the point of Zeno's paradoxes.

        I will refer to a story that is reported by some ancient Greek philosophers. Zeno, most do not know, was an advisor to Pericles, the man who established Democracy. One day he was giving a speech in the central Agora of Athens, trying to convince people that motion is impossible. When he stated his arguments, the philosopher Antiphon - a real person by the way - got up from his marble sit and started walking up and down in front of Zeno in a silent protest. Everyone laughed. However, by the end of his talk, it was reported that Zeno had convinced everybody in the audience that motion is impossible.

        The point is, we observe something we call motion but is this motion in 3-dimensional infinitely divisible space? This is the issue. Sure, we got motion; it is all over the place. But maybe it is not what people think it is. Maybe it is not motion in 3-D space but something like recreation of 3-D space from a higher dimensionality space, a sort of virtual reality.

        This is the issue. I think it is a misunderstanding that Zeno said motion is impossible. He specifically limited his argument to infinitely divisible 3-D space with absolute time. In Relativity for example, motion is possible because everything is in eternal motion in a 4-D spacetime, as I attempt to describe in the paper.

        Thanks again.

        Efthimios

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          • Post #33

          Dear Efthimios Harokopos,

          Thank you for replying. I see now that I left my point unclear:

          Me: "I assume that you mean that uncertainty frees us from a digital universe."

          Your response: "As a matter of fact I state the opposite that uncertainty is a feature of the digital world and determinism of the analog."

          My point is this: A digital world lacks connection. Since it functions in a cooperative manner, there must be some form of continuity. I presumed that you considered uncertainty to fill in gaps and, in effect, smear a digital nature so that it might connect enough to mimic continuity.

          Your analysis of this would be greatly appreciated. Please be as direct as necessary to make your point. Directness helps me to understand. Thank you.

          James

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          • Post #34

          Efthimios,

          I appreciate enormously your clarification of Zeno's point, because I am reminded of sitting through a conference presentation a few years ago, in which the presenter claimed to have solved Zeno's paradoxes (specifically, tortoise and hare; and arrow paradox) through some assumptions about time and space he had manipulated.

          At the Q & A following, I asked (innocently, in fact), "Well, is motion possible?"

          The reaction was as if I had two heads. Hadn't I just heard the presentation? The presenter and others went through all the main points of discussion about time and space. I asked again, "Then if the paradox is resolved, what's the answer: Is motion possible?" The time-space explanation took off again, with the added suggestion that perhaps I wasn't asking a proper question.

          I replied, "It's the question that Zeno asked."

          And I think that had never even crossed the presenter's mind, as he presented a solution to a problem that had never actually been posed.

          Zeno's question is equivalent to the origin of inertia. We still don't know.

          Tom

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          • Post #35

          There is an interesting story in one of Richard Ferynmans popular lectures on science. A student oncer asked him if, when he was viewing an object, if he was really 'seeing' the object or the light that is reflected from the object.

          He just replied by telling the story of a philophers who slowly starved to death because every time he was presented with a meal, he spent all his time contemplating whether or not the food was really just reflections of light.

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          • Post #36

          Dear Efthimios

          I read the response to Lieu's paper you cited above. Again the assumption is that quantum foam is a reality. This idea is speculative and is based on Born's probability interpretation. In my Beautiful Universe paper on which my present fqxi paper is based I have suggested that on the contrary nature may be precisely local, causal and deterministic at the minutest scale - and still produce quantum effects including probability. I also suggested that the Planck scale itself may be a fiction or much too tiny: G is determined by macroscopic experiments. It may well be that at the granular, ether (or whatever you call it) scale, its value is quite different.

          Concerning your question "If spacetime is granular, then what is there between the grains?"I would say it may be impossible to determine the physical nature of the granularity or any other hidden large dimensions the granules may reside in. We are talking about the stuff that makes stuff so we cannot project (what is the inverse of 'project'?) or macroscopic notions onto the granules of the universe. It will be sufficient to presume they have certain qualities (i.e. in my theory the lattice nodes have angular momentum, density, polarity, etc}.

          You said " disproving the granularity of space is equivalent to preserving the autonomy of the world. In my opinion, it is now too late for that." Can you please explain that interesting statement? thanks.

          Dear Ray et al,

          Concerning Zeno's paradox of divisibility and motion - in an ordered universal lattice where motion occurs by momentum transfer from node to node (as in my theory) , such questions will have obvious answers and will no longer pose any logical difficulties. There may be a message spelled out with the ether granules: "reductionism stops here!".

          Dear Anonymous

          The story of Feynman telling the story of the philosopher (who contemplated the reality of food and the light from it) is typical of his cavalier attitude to foundational questions. Feynman seems - perhaps wisely as far as his great work was concerned - to have adopted a pragmatic stance to unanswered quantum puzzles, using ad-hoc solutions and mathematical formulations even if they were not derived from more basic concepts. By the way a simple phrase like "just reflections of light?" was the foundational question of the 10th century answered for all time by the father of the scientific method, Al-Hassan Ibn Al-Haytham (Hazen) by his meticulous experiments with the camera obscura and their logical analysis in his book Kitab Al-Manazir. Before him it was believed (after Aristotle, I think) that we see because the eye projects visual rays onto an object. Hazen's book, translated from Arabic into Latin influenced the Renaissance and modern science. Moral of the story: foundational questions have to be answered eventually and not swept under the carpet, Feynman-style!!

          Good luck to us all! Vladimir

          Dear Ioannis,

          In my paper, I framed no hypothesis about the specific structure of reality, whether analog or granular. As Newton said, "hypotheses that are not deduced from the phenomena, whether mechanical or occult, have no place in science, especially in experimental physics"

          My essay was not about the ontology of spacetime. It was an effort to find a way of falsifying or corroborating the virtual reality conjecture. A virtual reality is generated by definition by a higher reality. I am not interested at all in specific proposals about possible ontologies. I began thinking this way long ago and soon I realized that unless those ontologies generate unique falsifiable prediction they are nothing more than metaphysical hypotheses.

          I am not sure I agree with your comment about not being able to manufacture a machine better than our brain. I certain respects I believe this has been accomplished many years ago. The human brain is very limited in mathematical operations but very fact in pattern recognition. In my experiment I am looking at a nanocomputer running a complex math algorithm. Computers are orders of magnitude better in doing math than people, I hope we agree to that. If you do not agree, try asking the smartest people in the world to matrix multiplication of solve partial differential equations with split boundary conditions.

          I cannot also discount the results of any experiments before they are performed.

          Thank you for your comments.

          E. Harokopos

          Hello James,

          I apologize for the delay in responding.

          You are raising important points. In my paper, I framed no hypothesis about the specific structure of reality, whether analog or granular. As Newton said, "hypotheses that are not deduced from the phenomena, whether mechanical or occult, have no place in science, especially in experimental physics"

          My essay was not about the ontology of spacetime. It was an effort to find a way of falsifying or corroborating the virtual reality conjecture. A virtual reality is generated by definition by a higher reality. I am not interested at all in specific proposals about possible ontologies. I began thinking this way long ago and soon I realized that unless those ontologies generate unique falsifiable prediction they are nothing more than metaphysical hypotheses.

          The problem with these hypotheses, as I said before, is that they hardly produce any new predictions on top of relativity and QM. They remain in the realms of metaphysics, at least for now. Thus, I avoid them. I concentrated in providing a short - due to space limitations - account to justify the virtual reality hypothesis and a possible experiment. As a scientist, I cannot go that far. I cannot speak about things I do not understand, things that are not deduced from the phenomena. I hope you will understand.

          However, I do not want to appear as if I am not escaping here. There are many possibilities I have thought of about this in the past. I think one that is plausible in the case that our reality is indeed generated by a higher reality is that the granular structure is part of a supersolid medium. The continuity part arises from the supersolid medium which allows for faster than light speeds in coordinating local operations in the virtual reality, in which the maximum speed is the speed of light in vacuum. The uncertainty arises from specific operations that I describe in another paper I am working on - actually a book - and I hope you will allow me to refrain from giving details here.

          Thank you.

          E. Harokopos

          13 days later

          Yiassou Efthimie,

          To the question, "is reality analog or digital" you answer "reality is fundamentally digital". To the same question, I respond that "we cannot know 'what is' but can only know what our 'measurements' of 'what is' are". In my paper "The Interaction of Measurement" I present a mathematical argument that shows we cannot know a quantity E(t) (as a function of time) directly through our measurements of E(t) - if these measurements involve an absorption of E in making the measurement.

          Although I have not included a discussion of this result in my essay, I thought you may be interested in considering it - especially as you are interested in the philosophy of science. But what I do include in my essay you will find very significant.

          The key result in my essay is to mathematically derive Planck's Law of blackbody radiation without using 'energy quanta' or statistics. This result shows that Planck's Law is an exact mathematical tautology that describes the interaction of measurement. This clearly explains why the blackbody spectrum obtained experimentally is so indistinguishable from the theoretical curve.

          I hope you can support my efforts to put this iconoclastic result in front of the 'panel of experts' for 'peer review'.

          Yia hara,

          Constantinos

          Dear Efthimios,

          Following my suggestion to possibly use moire effects to design an experiment to discover the granularity of the Universe: I just read a recent Nature journal article Vol 469 pp. 39-41 and pp 72-75 has an article by M. O'Sullivan et al entitled "Vernier templating..." that might prove interesting. The Vernier effect relies on two scales one slightly smaller than the other to measure extremely small distances, using a geometry identical to the moire effect. The paper studies a spherical configuration, and may provide some food for thought.

          Best wishes, Vladimir

          11 days later

          Dear Efthimios,

          Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

          Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

          Best wishes,

          Alan

            Dear Alan,

            Thank you very much for your kind words.

            Newton's law of universal gravitation, to which you referred, is a mathematical expression that has no physical meaning, according to his inventor, Newton himself. This is the well-known "hypotheses non fingo" phrase of Newton:

            http://en.wikipedia.org/wiki/Hypotheses_non_fingo

            The expression "fabric of spacetime", if I am not mistaken, is used for the 4-dimensional geometry of the spacetime of general relativity. There, gravity is not a force but only motion that results from the geometry of spacetime.

            I'm not sure we can talk about the mechanism of gravity when we do not know what gravity is in the first place. Regardless, as far as I know, any mechanism that has been proposed using some kind of "material influence", like graviton flux, for example, should create some type of drag on celestial bodies that has not been observed. It can also create heat that has not been observed.

            Gravity is a mysterious force. I believe, for whatever it worth, that what we call gravity is an "outside force" that makes our reality "functional", the term I use in my paper. The best mathematical model we have of this phenomenon today of that of general relativity on a macroscale.

            I am not sure if I answered your question, probably not I guess, but I am inclined to think that we should not be looking for a mechanism of gravity in our world of phenomena because it does not belong there. If it did, we would have known it already.

            Regards.

            Dear Efthimios,

            thanks for the link btw, I didn't know that one. I don't quite understand the "drag" aspect you talk of, other than the force of attraction. Why would a graviton create "heat"? okay through mechanical friction...ummm...I don't think we can say that this "hasn't been observed" due to no-one having entertained the idea in the first place.

            I just thought last night that the original quandry w.r.t the orbit of Mercury can likely be explained by the 'inclination hypothesis' i.e. that gravity is stronger on the plane of rotation of a celestial body. It's too much to explain in one go but there's plenty of circumstancial evidence to support this claim. I'm working on it right now.

            Best wishes,

            Alan