Juan,

I beg to disagree with you on a number of points.

First, I completely disagree with your claim that simply because c is a constant, Dx/Dt must trivially go to dx/dt. This is true *mathematically.* My point is that it makes no sense *physically.* I think I made it fairly clear *why* this makes no sense in my essay.

Now, before addressing your next points, let me first address your comments concerning quantum mechanics and quantum field theory. You are taking the difference between the two as being a contradiction. But two things can be different and not contradict. More to the point, quantum mechanics treats time as absolute (i.e. it ignores time) much as Newtonian mechanics (NM) does and thus both are generally interested in obtaining information about positions as functions of time. In the corresponding relativistic extensions of both QM and NM, space and time are now considered together and thus we are interested in functions of position and time (together). This does not, however, mean they contradict each other. In fact, as an example, it is a rather simple affair to derive NM from GR (see for example Shutz or Misner, Thorne, & Wheeler). They can't contradict if one can be derived from the other. The uncertainty relations still hold in QFT (in fact they are sometimes invoked in order to "explain" the spontaneous pair creation).

Now, regarding the points you made regarding discreteness (atomism, or whatever), let me start by quoting from Griffiths: "In principle, the force of impact between a bat and a baseball is nothing but the combined interaction of the quarks and leptons in one with the quarks and leptons in the other." So, ultimately, our classical interactions like that between a baseball and a bat, are really the sum of a bunch of quantum interactions. As you yourself just said, "there exist limits where that discreteness is indistinguishable from a continuum." Precisely my point! That limit is the macroscopic realm of classical physics! Classical physics works because we don't look closely enough or don't care for an increase in precision! But as soon as we do, we run into discreteness. Imagine you're an engineer making a speedometer for a car. Your boss asks you to make this speedometer more precise - say to 2 decimal places. Then he/she comes back and asks you to make it accurate to 4 decimal places. Then he/she wants 6 decimal places, etc. Eventually, though you're measuring a classical value, you're going to run into a *physical* - perhaps engineering is a better term - problem of *how* to get that information from the universe! The most accurate machines are discrete! In fact, the most accurate physical theory ever developed, i.e. in which theory comes closest to experiment, is QED *which is ultimately a discrete theory!*

As for the atomic chemists, I disagree, but then I note that the difference between your view and mine is simply a matter of interpretation. I have done a lot of work on the history of science and have reached a different conclusion. But then again, I know a lot of people who disagree with Thomas Kuhn's take on the history of science (including myself) and yet others who staunchly defend him. It's hard to be "right" when talking about the history of science in such a way.

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  • Post #24

Hello,

Interesting work but I do not agree with some of your key points.

I did not fully understand your position about Zeno's paradoxes. I also agree that we have all the mathematical answers. But I sense you tried to avoid answering directly whether supertasks are possible in nature.

Finally, I do not agree with the following statement:

"This would seem to imply that epistemic states are ultimately discrete on some level: our knowledge of the universe is discontinuous."

It is my understanding that this essay contest deals with the ontology of spacetime, not our epistemic states. These states are modified as science progresses and new experiments are performed.

and finally I do not also agree with this statement:

"Classical physics, with its inherent continuity, is nothing more than a convenient myth."

Einstein's Relativity is a continuous theory and actually the most successful of all times. General Relativity converges to classical Newton's Laws at the weak field limit. I do not see this theory and its continuity as a myth. I think the myth is "it from bit". This is what we should be targeting, in my opinion of course.

    Juan,

    Let me add a few other points that just occurred to me.

    I want to make it clear that it is entirely possible the universe is continuous. I'm arguing that we can't get truly continuous *knowledge* about it.

    Now, my view is perhaps colored a bit by my experiences. Oddly enough, I have degrees in mathematics, physics, and engineering and was a practicing engineer for awhile before entering academia. I think it this strange combination of all three that has given me the opinion that I have. While continuous measurements sound plausible, the engineer in me wants to know how in the heck we can make something that can truly measure something continuously.

    It goes back to that speedometer example again. Ask yourself this: whenever someone needs a truly accurate measure of speed, do they use an analog or a digital speedometer? How about a bathroom scale? If your response to the latter is to say we could use a traditional scale balance, I ask you what we used to accurately measure the weights we use on the balance? Another balance? And how accurate are those weights when compared with the *current* international standards for mass and local measurements of g? And I could go on asking questions like these and eventually you'd have to cite something that was quantum mechanical! Do you see my point?

    Actually, the most successful theory of all time, as measured by how closely it matches experiment, is QED.

    "It is my understanding that this essay contest deals with the ontology of spacetime, not our epistemic states. These states are modified as science progresses and new experiments are performed."

    Actually, this essay contest deals with reality. Reality is more than merely spacetime. Regardless, the point of my essay is that the epistemic states through which we access the ontology of spacetime (reality, whatever) necessarily limit the amount of knowledge we can obtain about the ontology of reality.

    One final point to add: mathematics (as with logic, linguistics, and computer science) is a formal science, i.e. one that follows from stated axioms. Formal sciences, by their very nature, are entirely self-consistent. The natural (or empirical) sciences (e.g. biology, physics, chemistry, etc.) are not necessarily self-consistent, e.g. quantum mechanics and general relativity don't quite mesh.

    Since Ian came to my defence on an earlier occasion, I'd like to say something here.

    Albert: you claim that "Einstein's Relativity is a continuous theory and actually the most successful of all times. General Relativity converges to classical Newton's Laws at the weak field limit. I do not see this theory and its continuity as a myth".

    Neither Ian (nor me) is saying that the theory is not continuous, but only that our knowledge of the world is discontinuous (and must always be so). This is also true of general relativity, and Einstein was aware of this. Indeed, it was just this lesson that led him away from the non-generally covariant field equations he had initially fixed on. The observable content, according to Einstein, was given by point-coincidences: only these respect the diffeomorphism invariance of the theory - the point was originally Erich Kretschmann's, but Einstein refashioned it.

    The consensus still remains that something like these relational point-coincidences exhaust what is observable in GR (Bergmann, Komar, Jim Anderson, Bryce DeWitt, and a host of others worked very hard on establishing this conclusion). So even if this essay competition is about the ontology of spacetime (which it isn't), we quickly stumble into epistemological terrain.

    Best,

    Dean

    Dear Ian Durham,

    Re: "Actually, the most successful theory of all time, as measured by how closely it matches experiment, is QED."

    Over 60 years of QED calculations of the anomalous magnetic moment [up to 12,000 Feynman diagrams involved in the latest such calculation] have produced the eight [or so] place accuracy of QED. Then, after this calculation is made, I believe the fine structure constant [upon which it is based] is adjusted, based on the results of the latest calculation of the anomaly. In my mind, this would lead, over 6 decades to a very accurate 'correlation' between these two.

    One issue that has bothered me is that, as of 1998, the vacuum energy, which is central to QED, was found to be overestimated by QED by 120 orders of magnitude. It would seem that this would call for 50 years of QED calculations to be redone, but I don't believe that this has happened. Am I missing some basic point here?

    Also, just a few years ago, the proton was assumed to contain a significant contribution from the virtual 'sea of strange quarks', but this has not turned out to be the case. I don't know whether to lay the blame for this at QED's door or QCD's door, but it would seem to be related to vacuum energy.

    What bother's me is that 'virtual particles' seem to be the best imaginable 'fudge factor' because the particles aren't measured [to my knowledge] but simply provide the means to 'fit' calculations to reality.

    And finally, the recent recent QED calculations of the proton radius based on the experimental data from 'muonic hydrogen' is off by 4 percent. Since this is the simplest possible system one would expect better of QED. Does this mean that QED now has one place accuracy? [Which would put it in the same realm as QCD.]

    I have generally been unable to get answers to these [and related] questions, and I wonder if you could help me understand what's going on.

    Edwin Eugene Klingman

    Dean,

    Thanks for the defense! Very well said (and considerably better than I could have done or was trying to do).

    Edwin,

    Oo, those are some excellent points you raise and I will confess to not having a proper answer at the moment. But I would agree that there are issues to be dealt with. My own inclination is that the problem lies in QCD or, rather, in the need for something beyond the Standard Model. But I'd have to think about it some more.

    Lev,

    Hmm. I like that. I'll have to read Collingwood.

    Ian,

    Excellent essay. Parts of it resonate considerably with the views in my essay. One of the most interesting points is where you write,

    "But what if the only way to get information about ontic states is through epistemic states? Further, what if the epistemic states themselves are discrete? How could we even determine if the underlying ontic states were continuous or not if the 'lens' through which we view them is discrete?"

    I would go even further to ask how we could even determine if the underlying ontic states exist at all. In what sense is it meaningful to talk of such ontic 'things in themselves' if we never have any direct access to them, even in principle? We are of course free to create speculative models to give coherence to our empirical observations (and it is the job of science to do so), but what is gained by the additional step of attributing a non-empirical ontic reality to a hypothetical 'somewhat' that such models supposedly describe?

    Thanks again for the fine essay.

    Regards,

    Tom

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      • Post #33

      @Ian,

      "Regardless, the point of my essay is that the epistemic states through which we access the ontology of spacetime (reality, whatever) necessarily limit the amount of knowledge we can obtain about the ontology of reality."

      Our epistemic states change constantly as technology progresses. Even until recently, we didn't know anything about fundamental particles, DNA, even the composition of close by planets. Thus, I view your argument as a distracting issue, a red herring.

      You have avoided dealing with the main issue, which is failure to unify discrete QM with analog GR.

      I get the feeling there is confusion in your argument between what we think we can know about reality and what it may turn out we can know about reality.

      In other words, you have not proved that what we can know is all we will ever know.

      @Dean

      "Neither Ian (nor me) is saying that the theory is not continuous, but only that our knowledge of the world is discontinuous (and must always be so)."

      In my opinion "discontinuous knowledge" does not imply that we cannot have knowledge about a continuous world. You have not proved that. Furthermore, I find universally quantified statements like "and must always be so" sort of dogmatic. Do you know what the future holds?

      I also disagree on a most fundamental level. There are many instruments physicists use that are purely analog in nature. Analog computers have been used for years to simulated dynamical systems. The knowledge those instruments provide is analog in this sense. Specific measurements may refer to discrete instants in time but those devices offer analog knowledge. Knowledge is not only the set of all discrete measurements we can get but also the means by which those measurements are obtained. Unless you can prove to me that the way we obtain knowledge is not knowledge in itself.

      Thus, I am afraid your argument is not even sound.

      Dear Ian Durham,

      The arguments that Edwin Eugene Klingman is giving for rebating your claim that QED is the most precisely tested theory are part of a more general argument given by me in my post of the day 16. I copy and paste:

      "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."

      Edwin Eugene Klingman recent remarks belong to my early points (i) and (iv). You did not reply to my remarks about QED, but I see in your recent reply to Edwin Eugene Klingman that you confess to not having a proper answer at the moment, which means that you have not answer to my points.

        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.

          A mistake in my post, the part where it says:

          "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."

          must be corrected to:

          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.

          > which means that you have not answer to my points.

          Indeed. I do not claim to know everything.

          I think you are still fundamentally missing my point, as Dean also made clear.

          I know full well that physicists believed in discrete electrons as early as the 19th century. Indeed, Newton proposed a particle theory of light even earlier! The question is, did they concurrently believe that *reality itself* was discrete? How about time? Or space for that matter? Believing that something *in* reality is discrete does not mean that one must believe reality *itself* is discrete.

          Regarding QFT versus QM, you are mistakenly equating "different" with "contradictory." QM and GR are contradictory in certain respects. QM and QFT are not. You are taking Dirac's comments entirely out of context.

          Regarding the speed of light as a constant, in fact it is merely a maximum value that can be obtained. It is known to be lower than its vacuum value in objects (e.g. water, crystals, etc.). Further, there have been suggestions (and some claim to evidentiary support) that it is variable under certain constraints (see work by Magueijo and others - I think I spelled his name wrong, but I don't have his papers handy).

          > You have avoided dealing with the main issue, which is failure to unify discrete QM with analog

          > GR.

          Since when was this the main issue? The contest rules said nothing of the sort (as I have pointed out before).

          Like Juan, I believe you are completely missing my point. Having read several other essays including Dean's, it is also apparent that I am not the only one making this same basic point.

          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.

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          • Post #41

          Dear Dr. Durham,

          Thank you for a truly enjoyable essay. And my sincerest condolences for your father-in-law.

          I do have a question. In my essay is a generalisation of the energy of a photon ('the Light'), which is that 'indefinable fusion' of the continuous and discrete, mentioned by de Broglie. Further, 'the Light' is the only 'quantum' theory that necessarily follows on from classical physics, and unequivocally demonstrates "Classical physics, with its inherent continuity, is nothing more than a convenient myth."

          Therefore, is 'the Light' ontic, epistemic, or both?

          All the best,

          Robert

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            • Post #42

            Ian,

            The thing I most appreciate about your essay -- and it's a great one -- is the recognition of the difference between mathematical continuity and the continuity of physical experience as enshrined in classical physics. Yes, it brings into question the very meaning of objective knowledge and its relation to ontology.

            I expect you'll continue to be the very model of a modern mathematician. Gilbert and Sullivan get no apologies from me. :-) I mean, putting aside what we're "not supposed" to look at closely in the way we see the world (Monet is a better example than Van Gogh, in my opinion), mathematics always deals with what lies beneath, as basic structure -- point and line and number. Yet the evolution of graphic art from flat images to perspectives incorporating a point at infinity anticipates the progress of mathematics incorporating that same image, in the abstract, on C*. If we need reasons to think that the ontology of mathematics connects with that of physics, we can always find them.

            I'm sorry for the loss of your father in law. Even though the event is inevitable, it seems that nothing prepares any of us for it.

            Good luck to you.

            Tom