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Dear Ian T. Durham,

I have loved your discussion of infinitesimals. Effectively, those exotic objects have evaded mathematicians during three centuries, giving a long controversy, which is still open!

You cite Robinson's work on "Non-standard analysis"; but, as you must know, this modern analysis has received criticism by other mathematicians. For instance, Connes is trying to obtain...

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Dear Ian T. Durham,

I have loved your discussion of infinitesimals. Effectively, those exotic objects have evaded mathematicians during three centuries, giving a long controversy, which is still open!

You cite Robinson's work on "Non-standard analysis"; but, as you must know, this modern analysis has received criticism by other mathematicians. For instance, Connes is trying to obtain a rationalization of infinitesimals using non-commutative geometry due to certain limitations of the non-standard analysis.

By a lack of space I did not discuss those interesting issues in my Essay. I met with the problem of the infinitesimals, when first tried to obtain a rigorous explanation for the neglect of second order corrections in the quanta n^(plusminus) in the canonical form, when deriving the results of classical physics. For instance, consider an elementary process describing the transport of energy between systems A and B (page 4 in my Essay). If the transport of energy is infinitesimal then the terms quadratic in epsilon are zero and one recovers the classical laws.

Time ago I named this "epsilon-calculus", although currently it is only a rule of thumb for our scientific applications and nothing that mathematicians would consider. Somehow as Max Planck used the concept of infinitesimal in his books in theoretical mechanics, although mathematicians considered his concept without mathematical meaning.

The problem is in finding an object epsilon different from zero but with (epsilon)^2 being zero. There is not real or complex number with those properties! Robinson want to characterize the infinitesimals using the new category of non-standard numbers, but if epsilon is a non-standard infinitesimal, then (epsilon)^2 is not zero, but a higher order infinitesimal. Another possibility could be the dual numbers and the Grassmann numbers, that have the property that their square is zero, but I have not studied this enough.

In practice, I merely take 'infinitesimals' to be very small real or complex numbers such that the squares are so small that cannot be measured in the lab. This is enough for practical applications and, at this point, I agree with you. However, time (fundamental time) is a different concept and a correct understanding of the (t --> t + dt) will need of a careful consideration of those mathematical issues.

I would like to comment the part where you discuss uncertainty relationships for light. You say that when Dt --> 0, "the time-energy uncertainty relation prevents us [...] from measuring the velocity of the object". But for a photon (Dx = c Dt) and the ratio Dx/Dt is well defined in the limit when Dt --> 0, giving the instantaneous speed of the photon. It is true that the relativistic uncertainty relations introduces a lower limit for t as a function of the uncertainty in momentum Dp, but the same limit is also introduced for x, and since the speed of the photon is a constant, both the average speed and the instantaneous speed coincide. In the classical limit the uncertainties go to zero (h --> 0), but we obtain the same speed: c.

Of course, we do not really measure what you call the "truly" instantaneous velocities, but neither we measure "truly" temperatures, "truly" electric currents, "truly" masses... For instance, suppose that the temperature of an object is T, when we place a thermometer in thermal contact the temperature of the system changes from (T --> T' = T + DT), where DT is the perturbation introduced by the thermometer. The goal is that if the thermometer is small enough when compared with the system size then T' will be near enough to T and can take T' as the temperature of the system. Indeed, one of the design goal of thermometers is to achieve the smallest possible size.

I fail to see why you consider that limitations in measurements imply that "our knowledge of the universe is discontinuous". Those limitations of our laboratories are with us since the very start of science, and all the classical physics, including its experimental branch has always been a science of the continuum.

You write "In fact it is doubtful, despite de Broglie's contention, that anyone prior to the twentieth century truly believed in a discontinuous universe, though they may have pondered the possibility". It is very difficult to accept that the chemists who developed the atomic theory of matter in the 19th century truly believed in a continuum universe. In a letter to Berzelius of 1812, Dalton Wrote: "The doctrine of definite proportions appears to me mysterious unless we adopt the atomic hypothesis".

You also write that the "results from the Wilkinson Microwave Anisotropy Probe (WMAP) have demonstrated that the geometry of the universe must be flat to better than 1%" and that "we of course have long known that it is locally curved". Well, we have also long known that spacetime is curved only in (geo)metric theories. In the so-named flat spacetime theories, e.g. the field theory of gravity (see the ref 19 in my Essay)), gravitation has a non-geometrical interpretation.

Moreover, this small deviation from flatness of less than 1% is the crux of the famous flatness problem in cosmology. Precisely, the non-geometrical approaches to gravity promise to solve this problem in a natural way (see e.g. [Nikolic]).

It seems that your "intuitive notion that causality is somehow related to continuity" is related to my emphasis on that fundamental time is a continuous quantity, unlike dimensional time, which can be discrete.

You write: "By quantizing fields we have seemingly turned something inherently continuous and non-localized into something discrete and localized". Precisely the quantum field theory suffers from the problem of localization, which obligated to physicists to introduce the concept of dummy spacetime. In quantum field theory, we no more can say where a particle "is" in spacetime. As emphasized by Sakurai in his well-known textbook: "It is important to note that the x and t that appear in the quantized field A(x,t) are not quantum-mechanical variables but just parameters on which the field operator depends. In particular, x and t should not be regarded as the space-time coordinates of the photon". See references 4-6 cited in my Essay for more technical details.

You continue with "To be clear, quantum electrodynamics, which is a quantum field theory, is the most accurate scientific theory ever developed, agreeing with experiment to within ten parts in a billion (10^−8)". The experimental support of quantum electrodynamics is excellent but it must be put in a right context. In the reference 6 in my Essay, I wrote: "Four main remarks may be done about the relativistic experiments and observations: (i) Precision tests of relativistic quantum electrodynamics are not normally carried out by directly comparing observations and experimental results to its theoretical predictions; (ii) the same tests are satisfied by formulations of relativistic quantum electrodynamics that are mutually incompatible between them; (iii) the experiments and observations only consider a very limited subset of phenomena; and (iv) both relativistic quantum electrodynamics and the relativistic quantum field theory are involved, at least indirectly, in some puzzling observations and glaring discrepancies". And then analyzed each remark by separate in the following two pages.

And then you add "But, ultimately, quantum field theory is built on quantum mechanics just as classical field theory is naturally consistent with classical mechanics". Well I opened the second section in my Essay, with a quote by Paul Dirac stating his dissatisfaction because quantum electrodynamics is not compatible with quantum mechanics. Several textbooks in quantum field theory emphasize some of the differences with quantum mechanics. In my Essay, I cited the standard textbook by Mandl and Shaw, but there is more.

You also write "Our only other recourse, then, is to assume that mathematical 'objects' have some kind of ontological status. The problem with this view is that there is no way to prove the ontological status of a mathematical object (one could always argue it is simply a representation of a physical object and is thus of a wholly different nature)". I think that would be good to emphasize here Feynman views in his celebrated course in Physics with Leighton, and Sands. They illustrated, in a marvelous form, the difference between physical reality and the mathematical objects used to represent them under certain conditions/approximations. One of their examples was about the difference between the physics of light and Euclidean geometry, which is very relevant to your own discussion of Euclidean geometry and radar guns.

In the last part of your Essay you write: "Classical physics, with its inherent continuity, is nothing more than a convenient myth. It's a nice approximation that works just fine when we don't look too closely". I think that this is a reflect of the traditional epistemological approach to physical reality, where science is perceived as a sequence of approximations to one supposed fundamental true.

Classical physics is not a myth, but a genuine branch of physics. From a purely theoretical point of view, classical physics would be considered a limiting case (h --> 0) of the underlying quantum physics. From an experimental point of view, classical physics is equivalent to quantum physics in those cases where the difference between both is less than the experimental error. In this modern epistemology, the word "approximation" would be used only for the cases where the difference is detectable.

[Nikolic] Some Remarks on a Nongeometrical Interpretation of Gravity and the Flatness Problem 1999: Gen. Rel. and Grav., 31(8), 1211-1217. Nikolic, Hrvoje.

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Dear Ian Durham,

The relation (Dx/Dt) = c = (dx/dt) has not only mathematical sense but physical meaning, because a measurement of the left hand side ratio (Dx/Dt) implies the measurement of the right hand side ratio (Dx/Dt), both ratios being equal to the speed of light. The issue of relativistic localization is studied with great detail in the reference 6 cited above in a previous post....

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Dear Ian Durham,

The relation (Dx/Dt) = c = (dx/dt) has not only mathematical sense but physical meaning, because a measurement of the left hand side ratio (Dx/Dt) implies the measurement of the right hand side ratio (Dx/Dt), both ratios being equal to the speed of light. The issue of relativistic localization is studied with great detail in the reference 6 cited above in a previous post. The physical explanation of why dx/dt can be measured can be obtained from the study done of dx and of dt. In your Essay you affirm that we cannot measure the instantaneous speed, because --as you repeat again above-- "Delta t must have a non-zero lower bound". However, when we repeat the analysis for x, we find that this lower bound for Dt does not prevent us from measuring the value of c, by the technical reasons stated in previous posts.

Regarding the relation between quantum mechanics and quantum field theory, you repeat some well-known trivial stuff, such as that time is absolute in quantum mechanics or that in relativistic quantum field theory we study functions of spacetime (x,t). However, nothing of these trivial stuff addresses the technical points stated in my previous posts, neither in the references cited in my own Essay. For instance, in my previous post, I remarked a very specific property of x where QM and QFT are clearly in contradiction, as is well-known, and even cited a standard textbook in QFT, where this contradiction is emphasized.

Dirac emphasized his disagreement because QED (QFT) was not compatible with QM. Dirac thoughts were given in my Essay and I partially quoted a relevant part in a previous post from mine. They key point here is that QFT does not reduce to QM, as Dirac stated and how has been rigorously proven in the references cited before. About other aspects of your post, the rest of references cited in my Essay give very detailed technical responses.

Effectively, as I have just emphasized "there exist limits where that discreteness is indistinguishable from a continuum", but this does not imply your claim that "classical physics is a myth". In fact, I am just saying the contrary than you! Classical physics is not myth, but a limiting case of quantum physics.

Your analysis of the car speedometer resembles the Schrödinger-cat paradox. This old paradox was based in a naive (without any mathematical rigor or experimental support) interpretation of the quantum theory, where a cat would be in some strange quantum superposition unless some physicist would look to it. But as our modern understanding of the quantum theory reflects, the cat will be classical even if no physicist is present at the lab. The same can be said about the car and the engineer. The car will be not classical or not according to engineer's boss choices about the number of decimal places in speedometer. With independence of the precision of the car's speedometer, this will be a classical car, not one in some weird quantum superposition between New York and Paris, for instance. The same argument hold for your appeal to bathroom scales; with independence of its precision, you will be not superposed between bathroom and kitchen.

You repeat your comments about QED being the most accurate physical theory ever developed, but again you omit the technical details. I already put your statement in a right context in my first post of day 16!

You add now that QED is "ultimately a discrete theory". But this is another exaggeration, because QED is a theory of quantum fields and the quantum fields are continuum objects (with continuum spectral decomposition and continuum basis). Moreover, QED also treats volume and others properties as a continuum.

You are right on the existence of different views regarding history. However, not all the views are in the same footing (for instance, the view that Einstein played no role in the development of relativity is maintained by some very few historians, but their view is strongly rejected by the rest of historians). Regarding the history of the atomic theory, it is broadly accepted by historians of science (and I do not know anyone who disagree!) that chemists of the 18th and 19th were the first to develop a scientific theory where the Universe was considered to be composed of tiny particles called atoms. I have given arguments, the names of relevant scientists as Dalton, and how using this discrete model, they were able to explain empirical laws then without any explanation.

Moreover, as is well-known to historians of science, electricity was considered to be made of discrete negative particles before the 20th century:

"Now the most startling result of Faraday's Law is perhaps this. If we accept the hypothesis that the elementary substances are composed of atoms, we cannot avoid concluding that electricity also, positive as well as negative, is divided into definite elementary portions which behave like atoms of electricity."

Those discrete units of electricity were named "electrons" by physicist Stoney in the 19th, who added about his paper "On the Physical Units of Nature" the following: "I called attention to this minimum quantity of electricity as one of three physical units, the absolute amounts of which are furnished to us by Nature, and which may be made the basis of a complete body of systematic units in which there shall be nothing arbitrary".

As a conclusion, the affirmation done in your Essay on that everyone before the 20th century believed in a continuum Universe is one without historical basis.

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Dear Ian Durham,

Dean Rickles replied to Albert, and mixing Albert's viewpoints (I do not agree with all that he said) with the mine (e.g. you name me in a reply to him) only can give up to further confusion. I will repeat my point.

I have said that there exist limits where discreteness is...

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Dear Ian Durham,

Dean Rickles replied to Albert, and mixing Albert's viewpoints (I do not agree with all that he said) with the mine (e.g. you name me in a reply to him) only can give up to further confusion. I will repeat my point.

I have said that there exist limits where discreteness is indistinguishable from a continuum, which implies that your claim that "classical physics is a myth" is a complete exaggeration. Classical physics is not myth, but a well-defined subset of physics. Classical physics is a scientific discipline not a myth!

I offered some technical details about my point that you have avoided. Following your example of a car speedometer, I emphasized why a car is a classical object even if you decide to "look closely enough" (in your own words). I also drew the interesting analogy between your claims and the confusion of the earlier physicists regarding the state of a cat (Schrödinger-cat paradox). The cat will be in a classical state, with independence of the observations of the physicists, if any observing the cat at all! Analogously, the car will not be in a superposition between Paris and New York even if you ask engineers for a better car's speedometer.

You say "I know full well that physicists believed in discrete electrons as early as the 19th century." This is again misleading by several reasons. Physicists and chemists developed a scientific theory of matter which contained a discrete unit of electricity, instead of merely believing in discreteness as some philosophers did; these scientists named "electrons" to those discrete units (your term "discrete electrons" is misleading because electrons are, by definition, discrete); they gave the value for this elementary unit of charge; and they emphasized its fundamental role for our understanding of the Universe. Stoney wrote in his paper "On the Physical Units of Nature":

"I called attention to this minimum quantity of electricity as one of three physical units, the absolute amounts of which are furnished to us by Nature, and which may be made the basis of a complete body of systematic units in which there shall be nothing arbitrary".

As a consequence, the claims made in your Essay have not historical basis. Personally, I find really curious that the discrete theories developed by physicists and chemists in the 18th and 19th suggest to you that they would believe that "reality itself" is continuous, whereas you affirmed above to us your belief on that QED is "ultimately a discrete theory", without being aware of that QED is a theory of fields and that the fields are continuum systems (the fields have a continuum spectra).

Regarding the relation between QFT and QM, you opine that I am "taking Dirac's comments entirely out of context". The problem here is not that you ignore what Dirac really said, but that you also ignore the additional technical details and the rigorous references given.

For instance, I remarked one specific contradiction between QFT and QM at least in three occasions, and your response has consisted on completely avoiding to comment about this contradiction the same number of times.

In previous posts I discussed how, contrary to misguided claims done in your Essay, the constant speed of light c can be measured. The fact that the phase speed v = c/n is less than c for materials as water, does not mean that we cannot measure c. You then move away from materials and introduce Magueijo cosmological model where the speed of light c is substituted by a variable speed c*. However, his work is highly speculative (I was discussing our current models based in the universal constant c and how we measure that c in the lab). Moreover, he introduces a larger c* for the early Universe, but maintains a value c for the present Universe. Finally, it must be emphasized that, even if his model was finally verified, it does not change my remark (Dx/Dt = c = dx/dt) about the constant c.

Finally, you add "Oh, and regarding my lack of 'technical details' surrounding the claims of the accuracy of QED, Google 'precision tests of QED' and read what comes up." Again you ignore what is being said. Nobody here doubts of that accuracy. I wrote "The experimental support of quantum electrodynamics is excellent". The part that you omit both in your Essay and in this forum is that accuracy may be put on the right context.

The right context is the following, as I wrote before: "Four main remarks may be done about the relativistic experiments and observations: (i) Precision tests of relativistic quantum electrodynamics are not normally carried out by directly comparing observations and experimental results to its theoretical predictions; (ii) the same tests are satisfied by formulations of relativistic quantum electrodynamics that are mutually incompatible between them; (iii) the experiments and observations only consider a very limited subset of phenomena; and (iv) both relativistic quantum electrodynamics and the relativistic quantum field theory are involved, at least indirectly, in some puzzling observations and glaring discrepancies". And then analyzed each remark by separate in the following two pages."

Edwin Eugene Klingman has done similar remarks about QED regarding points (i) and (iv), you replied to him confessing to not having a proper answer, you replied me saying more of the same... but now without offering any technical response to any of us, you go up and ask us to read Google hits for "precision tests of QED"! I am sorry to reveal you that this material was already known to us and that, evidently, it does not address the points (i)-(iv).

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Dear Ian,

I onjoyed reading your essay. Several of the issue that you discuss have been paradoxical for ages and have not been resolved satisfactory. There is, however, a way to get around those issues, which I will explain below by addressing them specifically.

1. Page 4. the second expression for the average speed makes the assumption that the limit can be determined, because dt...

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Dear Ian,

I onjoyed reading your essay. Several of the issue that you discuss have been paradoxical for ages and have not been resolved satisfactory. There is, however, a way to get around those issues, which I will explain below by addressing them specifically.

1. Page 4. the second expression for the average speed makes the assumption that the limit can be determined, because dt (read d as Greek delta) can be made arbitrary small. This is not the case. As I describe in my essay, for electrons, dt has a lower bound equal to 10 exp (-20) sec. A The electron would cease its existence by 'making it smaller'. In other words, mathematically, one can determine the stated limit, but it is unphysical. The velocity of a massive particle is well-defined, even although dt is different from zero. The velocity of a particle does not need to be described in terms of a mathematical limit at all: a definition in terms of a discrete ratio is sufficient, i.e. v = dx/dt, where dx and dt are constrained by quantum condition (4) stated in my essay. In this way, an electron can be assigned a velocity without measuring this (see page 6 of your essay where you discuss the issues around this).

2. Page 5. Classical Light. I assume you mean light described in terms of classical theory.

3. Page 5. "by Brukner and Zeilinger to argue that the continuum is nothing but a mathematical construct, a view I wholeheartedly endorse". I do not necessarily agree with this view. As I describe in my essay, continuity needs to co-exist with discreteness. In the theory I describe, two underlying continuous fundamental fields are needed to explain the existence of particles, interaction between particles, and dynamically emergence of local discrete space and time.

4. Page 5. "So what happens in the limit as dt --> 0 for classical light?" As I indicated under 1., dt cannot be smaller than 10 exp (-20) sec. Light cannot be attributed a discrete time, unless one wants to define it as wavelength/c. The latter is not very useful, since the time would be wavelength dependent.

5. Page 5. "Suppose we decrease dt while leaving dx unchanged. As dt gets smaller and smaller, it implies we are measuring the difference between x1 and x2 more and more rapidly. Lest we forget, classical physics limits how rapidly information can propagate. At some point, without changing dx, we will be empirically prevented from further reducing
dt since the ratio of
dx to dt cannot exceed the speed of light. So, if we wish to take
dt --> 0, we must take dx--> 0 in order to keep the ratio at or below the speed of light."

There is an implicit assumption made that dt can be made arbitrary small, which is not the case as I explained earlier. For a stationary electron dx=0, while dt=10 exp (-20) sec, such that the ratio dx/dt=0 and there is no issue with violating the speed-of-light as one would get by assuming that dt can be made arbitrary small. When the speed of an electron increases, both dx and dt increase, but their ratio cannot exceed the speed-of-light c. In my essay, I explain that this is due to the fact that the internal speed of random spatial motion of an electron is equal to the speed-of-light. The details can be found in the second report on my website.

6. Page 5. "The classical theory of light assumes light is a wave which is an inherently non-local phenomenon". Indeed. This result in a paradoxical behavior. However, is is also known that light consists of photons, which propagate at the speed-of-light, and posses a kind of corpuscular behavior when detected. When one assumes that photons are oscillating blobs, then they do not behave as a classical wave. Still, a 'wavelength' can be assigned, which is equal to the length of the oscillation.

I have a few more comments, which I may write up later.

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Dear Ian Durham,

I found some errors in this essay:

You write: ''Any interaction necessarily requires an exchange of information''.

It is an erroneous statement; I can show you an example of interaction without any exchange of information. The Black Hole's event horizon is a boundary in spacetime through which matter and light can only pass inward towards the mass of the black...

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Dear Ian Durham,

I found some errors in this essay:

You write: ''Any interaction necessarily requires an exchange of information''.

It is an erroneous statement; I can show you an example of interaction without any exchange of information. The Black Hole's event horizon is a boundary in spacetime through which matter and light can only pass inward towards the mass of the black hole. Nothing, not even light, can escape from inside the event horizon. Therefore, if a body cross the event horizon, the exchange of information between the Black Hole and falling body is not possible because information can flow in one direction only. Thus, this statement is wrong.

You write: ''This idea simply formalizes the somewhat intuitive notion that causality is somehow related to continuity. If spacetime is discontinuous, how do we know that this information couldn't jump around' from point to point? Continuity guarantees that the information follows a nice, orderly 'path' between A and B. This should make it easy to see the conceptual attraction of a continuous reality''.

1) In quantum mechanics particles don't follow a nice, orderly 'path' between A and B, and the position of a free particle is uncertain - so we can consider space as discontinuous. In spite of fact that particles don't follow a nice, orderly 'path' between A and B and all macroscopic bodies are made of quantum particles, we do not observe any violation of causality in our everyday life or macroscopic experiments. Thus, your statement is wrong: even if the space and behavior of the particles is discontinuous, the causality holds in our everyday life.

You write: ''Now, any attempt to measure velocity (or just about any other physical quantity, for that matter) requires an interaction between the observer and the system under observation (sorry folks, there's just no way around that)''.

It is an erroneous statement; we can know the velocity of a particle (or even a macroscopic body) without measurement/interaction. Imagine the radioactive decay of an atomic nucleus when two its clusters/parts fly in opposite directions. If we measure the energy (mass and velocity) of one cluster/part, we'll know the mass and velocity of another cluster without measurement (interaction) with this object.

Another example - we can know the velocity of distant star without interacting with one by observing its motion concerning other stars. in fact, we do not interact with this star but we know its velocity.

You write: ''Though the universe itself has cleverly prevented us from determining whether or not it is continuous, I'd like to believe that it is''

It is an erroneous statement; we can determine whether or not the Universe is continuous by observing the expansion of the Universe: If the distance between galaxy clusters is increasing today, everything must have been closer together in the past. It means that in the first microseconds of expansion the Universe was very small and therefore finite in volume. Since the Universe has a finite age, the Universe will have the finite volume always, in spite of expansion. Since the Universe has a finite volume, it must have the edges (holes), because all objects with finite volumes have borders. And the space with holes is discontinuous because a hole is the absence of spacetime. Thus, since the Universe is expanding, therefore it must be finite and discontinuous. Do you see any flaws in this reasoning? Hence, the Universe is discontinuous but not continuous.

Sincerely,

Constantin

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Dear Sir,

Your examples of ontic state may be correct to some extent (subject to variations of its mechanical functioning), but that of epistemic state is not correct for the simple reason that knowledge is not probabilistic. Knowledge is the result of measurement and it is the same as wave-function collapse, which freezes the state. Hence, as Spekkens points out, it is no longer...

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Dear Sir,

Your examples of ontic state may be correct to some extent (subject to variations of its mechanical functioning), but that of epistemic state is not correct for the simple reason that knowledge is not probabilistic. Knowledge is the result of measurement and it is the same as wave-function collapse, which freezes the state. Hence, as Spekkens points out, it is no longer probabilistic. We have discussed this point elaborately in our essay.

We have defined reality precisely and shown that Nature is mathematical only in specified ways. Berkeley’s so-called paradox is not a paradox at all if you view in the right context. The original equation represented the curvature of a parabola and the relationship between its projections along x and y axes respectively. The small increase in y will lead to corresponding increase in the value for x. Till such time, the equations are alright. But the problem arises when we start manipulating the figures out of context. In the present context it becomes physically meaningless. For example, if the cost of 5 bikes equals the cost of a car, the cost of 1 bike equals the cost of 1/5 of a car. But what is 1/5 of a car? It is a meaningless statement. These meaningless manipulations lie at the heart of the present problem.

The simplest answer to Zeno’s paradox is that velocity is related to the mass of the body that is moving, the energy used (force applied) to move it and the total density of and the totality of the energy operating on the field. These are all mobile units against the back drop of the field that is static with reference to these. Middle of the distance is related to the frame of reference, which is relatively static with reference to the other mobile aspects. Thus, it is like comparing position and momentum. They do not commute. Hence there is no paradox, which is borne out of experience. While the middle of the distance is gradually reduced, the velocity is not reduced by the same proportion.

Ever since Newton propagated his second law, acceleration has been highly misunderstood by the scientific community. Before we give a proper explanation for the mechanism of acceleration, let us analyze the equation F = ma.

Without any qualifying word, F here is to be understood as any impressed force. The function of a force is to displace bodies from their position. The force can be impressed by a source only. After the force is impressed, the body is displaced. Thereafter, its contact with the source is cut off. Now the body moves with inertia, which remains constant in the absence of any other force. Thus, the equation should have been F = mv.

There may be occasions where the source impressing the force moves with the body. One example is an engine pushing a train or a cab. Here after the initial displacement, inertia takes over. But, the friction with the rail or the road retards the velocity. The force, which is moving in the same direction, again comes in contact with the body and again pushes it. This leads to a continuous change of velocity, the rate of which is called acceleration. But as can be seen, another force of friction is acting to generate acceleration, which has not been included in the equation. Thus, the mathematical form of Newton’s second law is wrong.

To understand the true nature of acceleration, we have to understand wave motion. According to the latest findings of LHC, the early universe was a ‘perfect fluid’, not an ‘explosion of gases’ that is the basis of all current theories. We posit that this fluid formed the primary field. Particles are subsequent generations of this field through confinement. A wave is a disturbance in a fluid medium where the particles transfer the momentum only. This implies that the particles in a field are displaced temporarily and due to inertia of restoration (elasticity), regain their position and are subjected to the same force. Since fluid mediums do not have a strong confinement like solids, each particle pushes the others over a field leading to a chain reaction, which goes on repeating. The pushed particle, which was at rest, pushes the first particle back canceling half of its impact and transferring the other half to the next particle. We call this motion as “kampa”. Since this transfer of energy involves over a field covering the amplitude of the wave and is further modified by the density (which is related to mass per unit volume) of the medium, the equation for momentum is ½ mv^2 at every point (most text books give a wrong explanation of this phenomenon).

Now, imagine a situation where the impressed force overcomes the inertia of restoration. The particle is displaced fully and in turn it displaces the next particle. There will be a reaction as above, but the rate of change of velocity will be reduced gradually. The particle will come to rest after sometime. Since the original particle will be going back to the source after sometime, the end particle will be subjected to a similar force in a chain repeatedly. We call this phenomenon “chiti”. This last particle in a “chiti” then acts as a center of mass for other interactions. This finally leads to the formation of a structure because, as we have explained earlier, all structures have a center of mass surrounded by the extra-nuclear field and confined by orbits.

A paper published in October, 2005 issue of Notices of American Mathematical Society shows that the same mathematics governs the theory of dynamical systems used to plan trajectories of space crafts and the theory of transition states of chemical reaction. The same laws of physics hold both for quantum world and the macro world. In our essay we had shown how the Uncertainty relation has been misinterpreted. It is not a law of Nature. It is a result of natural laws relating to observation that reveal a kind of granularity at certain levels of existence that is related to causality. The left hand side of equations represents free-will, as we are free to choose the parameters. The right hand side represents determinism as the outcome is based on the input in predictable ways. The equality sign prescribes the special conditions to be observed. There is no need to complicate the issue.

In your example, if you decrease delta t while leaving delta x unchanged, what it means is that the object is stationary in the frame of reference as time passes. As delta t gets smaller and smaller, it does not imply that we are measuring the difference between x1 and x2 more and more rapidly, because the concept of x1 and x2 has simply vanished. Thus, the rest of your examples are a wrong description, hence not valid.

When you say: “epistemic states are ultimately discrete on some level: our knowledge of the universe is discontinuous”, what it really means is that we have incomplete information. Knowledge is related to unification of the various sensory impulses to create a stable memory. None of the fundamental forces of Nature in isolation is useful for creation. Only collectively they can create stable systems. Similarly, knowledge, which unifies the different perceptions, is stable and continuous. But depending upon individual perceptions, it may have limited information.

Both space and time are related to the order of arrangement in the field, i.e., sequence of objects and events contained in them like the design on a fabric. Both space and time co-exist like the fabric and its back ground color. The perception of this sequence is interrupted by an interval however infinitesimal. The interval between objects is called space and that between events is called time. We take a fairly intelligible and repetitive interval and use it as the unit, where necessary by subdividing it. We compare the interval with this unit interval and call the result measurement of space and time respectively.

Since space and time have no physical existence like particles and fields, we use alternative symbolism of objects and events to describe them. Thus, what Euclid called space is not the interval between objects, but the basic frame of reference on which the objects are placed as markers. To this extent he is right. Dedekind and others did not know this concept. Hence they wrongly held that “it is possible to construct discontinuous spaces in which Euclidean geometry holds”. Geometry is related to measurement of space and no measurement except distance (line) is possible in discontinuous spaces like in the interval between a point on Earth and another point on the Sun or Moon. However, this fallacy was not apparent to the others who built theories upon such invalid foundation. Since space is the interval between objects, the space is continuous throughout the Universe.

What the Wilkinson Microwave Anisotropy Probe (WMAP) have demonstrated is that the field encompassing everything in the Universe is the same.

Causality and determinism are two sides of the same coin. The left hand side of equations represents free-will, as we are free to choose the parameters. The right hand side represents determinism as the outcome is based on the input in predictable ways. The equality sign prescribes the special conditions to be observed. There is no need to complicate the issue. The direct relationship between causality on a given space-time and the continuity of the Lorentzian distance on that space-time is only apparent to the observer and not real to the systems being observed. Information or knowledge is related to observation by the observer. It may or may not represent the true state of the system being observed.

Special Relativity is not only conceptually, but also mathematically wrong. This is what Einstein describes in his 30-06-1905 paper "On the Electrodynamics of Moving Bodies":

Einstein: We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:

1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.

2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Our comments: Here clock at A is the privileged frame of reference. Yet, he tells the opposite by denying any privileged frame of reference. Further, his description of the length measurement is faulty. Here we quote from his paper and offer our views.

Einstein: Let there be given a stationary rigid rod; and let its length be l as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:-

(a) The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.

(b) By means of stationary clocks set up in the stationary system and synchronizing in accordance with §1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated “the length of the rod”.

In accordance with the principle of relativity the length to be discovered by the operation (a) - we will call it the length of the rod in the moving system - must be equal to the length l of the stationary rod.

The length to be discovered by the operation (b) we will call “the length of the (moving) rod in the stationary system”. This we shall determine on the basis of our two principles, and we shall find that it differs from l.

Our comments: The method described at (b) is impossible to measure by the principles described by Einstein himself. Elsewhere he has described two frames: one fixed and one moving along it. First the length of the moving rod is measured in the stationary system against the backdrop of the fixed frame and then the length is measured at a different epoch in a similar way in units of velocity of light. We can do this only in two ways, out of which one is the same as (a). Alternatively, we take a photograph of the rod against the backdrop of the fixed frame and then measure its length in units of velocity of light or any other unit. But the picture will not give a correct reading due to two reasons:

• If the length of the rod is small or velocity is small, then length contraction will not be perceptible according to the formula given by Einstein.

• If the length of the rod is big or velocity is comparable to that of light, then light from different points of the rod will take different times to reach the camera and the picture we get will be distorted due to the Doppler shift of different points of the rod. Thus, there is only one way of measuring the length of the rod as in (a).

Here we are reminded of an anecdote related to Sir Arthur Eddington. Once he directed two of his students to measure the wave-length of light precisely. Both students returned with different results – one resembling the accepted value and the other different. Upon enquiry, the student replied that he had also come up with the same result as the other, but since everything including the Earth and the scale on it is moving, he applied length contraction to the scale treating Betelgeuse as a reference point. This changed the result. Eddington told him to follow the operation as at (a) above and recalculate the wave-length of light again without any reference to Betelgeuse. After sometime, both the students returned to tell that the wave-length of light is infinite. To a surprised Eddington they explained that since the scale is moving with light, its length would shrink to zero. Hence it will require an infinite number of scales to measure the wave-length of light.

Some scientists try to overcome this difficulty by pointing out that length contraction occurs only in the direction of travel. If we hold the rod in a transverse direction to the direction of travel, then there will be no length contraction for the rod. But we fail to understand how the length can be measured by holding it in a transverse direction to the direction of travel. If the light path is also transverse to the direction of motion, then the terms c+v and c-v vanish from the equation making the entire theory redundant. If the observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod while moving with it, he will not find any difference what-so-ever. Thus, the views of Einstein are contrary to observation.

His “mathematics” using the equation for the sphere is all wrong. For example, he has used equations x2+y2+z2-c2t2 = 0 and ξ2 + η2 + ζ2 - c2 τ2 = 0 to describe two spheres that the observers see of the evolution of the same light pulse. Apart from the fact that the above equation of the sphere is mathematically wrong (it describes a sphere with the center at origin, whose z-axis is zero, i.e., not a sphere, but a circle), it also shows how the same treats time differently. Since general equation of sphere is supposed to be x2+y2+z2+Dx+Ey+Fz+G = 0, both the equations can at best describe two spheres with origin at (0,0,0) and the points (x,y,z) and (ξ, η, ζ ) on the circumference of the respective spheres. Since the second person is moving away from the origin, the second equation is not applicable in his case. Assuming he sees the same sphere, he should know its origin (because he has already seen it, otherwise he will not know that it is the same light pulse. In the later case there is no way to correlate both pulses) and its present location. In other words, he will measure the same radius as the other person, implying: c2t2 = c2 τ2 or t = τ.

Again, if x2+y2+z2-c2t2 = x’2+y’2+z’2-c2 τ 2, t ≠ τ.

This creates a contradiction, which invalidates his mathematics.

Through all this, you have not defined what reality is. We suggest you read our essay.

Regards,

basudeba.

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Dear Sir,

We were following your views on gravity and Coulomb's law. Here is our comment on that.

Before we discuss whether the force we were referring to was gravity, we will like to discuss something about force itself. A force is experienced only in a field (we call it rayi). Thus, it is a conjugate of the field. If something is placed in a field, it experiences something else....

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Dear Sir,

We were following your views on gravity and Coulomb's law. Here is our comment on that.

Before we discuss whether the force we were referring to was gravity, we will like to discuss something about force itself. A force is experienced only in a field (we call it rayi). Thus, it is a conjugate of the field. If something is placed in a field, it experiences something else. This something else is a kind of force. Depending upon the density variations of the field, we experience the force differently. Hence we call it by different names. While the field is one, the forces are many. Since they are conjugates, we can also say that different forces create different variations in the field.

The basic nature of the field is equilibrium. The basic nature of forces is displacement. This gives rise to two different types of inertia: inertia of motion due to forces and inertia of restoration (elasticity) due to the field. This leads to both these inertia acting against a point of equilibrium. In such a scenario, the combined effect leads to confinement around the point of equilibrium. The confined structure is called particle. Thus, all particles have a central point of mass or nucleus, an extra nuclear field surrounding it and fixed orbitals confining it. This is the common feature of all particles be they quarks or the Cosmos. The confinement may also cover the field without the central point. This is caused due to non-linear interaction of the forces. We will describe the mechanism separately. In such a case the field behaves like a fluid. The latest finding of LHC is that the Universe was created from such a super-fluid and not gases. The confined field also interacts with the Universal field due to difference in density. This in turn modifies the nature of interactions at different points in the medium (Universal field).

A force can act only between two particles as only a particle can influence the field, which in turn can be experienced by another particle. If the external force of the field is more than the confining force of the two particles, then the two particles break up and join to form a new particle. We call this “sambhuti”. In the opposite case, the two particles experience the force without being internally affected. The force acts between the centers’ of mass of each treating each as a point particle. We call it “bibhuti”. This second category of relationship, which we call “udyaama”, is known as gravity. Since it stabilizes the two bodies at the maximum permissible distance between them depending upon their respective masses, we call it “urugaaya pratisthaa”. For reasons to be discussed separately, this is possible only if gravity is treated as a composite force.

The first category of forces, which are interactions between two bodies, acts differently based on proximity-proximity, proximity-distance, distance – proximity and distance – distance variables. We call these relationships “antaryaama”, “vahiryaama”, “upayaama” and “yaatayaama” respectively. This interaction affects the field also inducing various local disturbances. These disturbances are known as “nitya gati”, “yagnya gati”, “samprasaada gati” and “saamparaaya gati” respectively. Any particle entering the field at those points feels these disturbances, which are known as the strong nuclear interaction, weak nuclear interaction, electromagnetic interaction and radioactive disintegration respectively. Thus, you can see that gravity belongs to a completely different group of forces and cannot be integrated with other fundamental forces of Nature in the normal process. Yet, it has a different function by which other forces can be derived from it. We will discuss that separately.

According to our theory, gravity is a composite force of seven forces that are generated based on their charge. Thus, they are related to charge interactions. But we do not accept Coulomb's law. We have a different theory for it. We derive it from fundamental principles.

According to our theory, all particles are locally confined fields. This confinement takes a three fold structure for the particle - center of mass or nucleus, extra-nuclear field and the confining orbitals. If we take into account the external field with which the particle interacts, it becomes a four-fold (3+1) structure. The particle interacts with the field in two ways. If the internal energy distribution cancels each other with a little inward pull, then it behaves as a stable particle. Thus, we have derived theoretically the charge of proton in electron units not +1, but +10/11. Similarly, the charge of neutron is not 0, but -1/11. This makes the atom slightly negatively charged. This excess negative charge is not experienced out side as it is directed towards nucleus. But it is released during fusion and fission.

The confinement described above takes place where the external field dominates to confine the particle. Here the particle becomes negatively charged. In the opposite case, the particle becomes positively charged. The particles are classified as positively charged or negatively charged according to whether the external field dominates over confinement or the confined force dominates over the local field. Since equilibrium is inherent in Nature, in either case, the particles search for their complements to become full. The less negative part of the proton (since it is +10/11, it has -1/11 negative charge) seeks to couple with the electron to become -1/11. This makes hydrogen atom highly reactionary.

The combined charge of proton and electron (-1/11) seeks the neutron since it has an equal charge. Thus, the opposites do not attract and same charge does not repel. It is not the opposite either. The charge interaction can be of four types:

positive + positive = explosive.

Positive + negative (total interaction) = internally creative (increased atomic number)

Positive + negative (partial interaction) = externally creative (becomes an ion)

Negative + negative = no reaction.

Regards,

basudeba.

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