In Search of Continuity: Thoughts of an Epistemic Empiricist by Ian Durham

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

Hi Ian, his declaration of universality or put simply "every object attracts every other object equally in all directions" is a BIG assumption which is then set in stone within his gravity equation. No wonder it can't be reconciled with particle based QM! Why did no-one at the time of Newton consider the Archimedes screw as a mechanical method for explaining the force of gravity, his spooky action at a distance?? The history of science would have been very different if someone had imo! Best wishes, Alan

Ian,

Having read again more carefully, please indulge me in correcting a slight inaccuracy that I think has been responsible for many misunderstandings of general relativity. On p. 7, you characterize spacetime in general relativity as being modeled on a four dimensional Lorentzian manifold. Not quite true. It is a four dimensional pseudo-Riemannian manifold of Lorentzian metric properties.

This is important, especially for the relativity-doubters (with which this forum abounds) because it explains classical gravity symmetry and time reversal symmetry. The manifold is pseudo-Riemannian because every Riemannian manifold is, in fact, orientable. The non-orientability of Lorentz invariance ("all physics is local") informs us of the relationship of "empty space" to matter -- it was Einstein's (and Mach's) desire to reduce physical epistemology to the properties of matter alone. Lorentz invariance with spacetime produced a model finite in time and unbounded in space.

Tom

My apologies. I did not know that "Lorentzian manifold" is a special case of the pseudo-Riemannian manifold until I looked it up. (I was unfamiliar with the term.) Other points re relativity still hold, nevertheless.

Tom

Hi Ian,

once again, you wrote an excellent Essay. In particular, I enjoyed with the "EVERY GOOD MYTH NEEDS AN ANCIENT GREEK".

Have my condolences for the death of Lawrence Brod.

Best regards,

Ch.

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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).

Dear Albert,

You write:

"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?"

My point was that given our means of engaging with the world is based on discrete events (which in our most mature theories it is), any knowledge we think we gain that goes beyond these discrete events will be inference (metaphysics, in fact). Unless we radically alter our means of engaging with the world (by measurement, using relations between properties) then this will indeed always be so: this strikes me as a generalization that can be made! (I'm not completely happy with the phrase "discontinuous knowledge" in any case, and was led to it by the nature of the discussion).

This relates to your second point. You write:

"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."

I showed in my own essay how the representation (be it digital or analogue) cannot be used as a guide to the nature of the target system being represented, since one can model one and the same target system using either method of representation. Further, I'm not sure what your argument is that gets you from an analogue instrument to analogue knowledge. Given that the analogue/digital distinction refers to representation, I'm not entirely sure what "analogue knowledge" could mean? Knowledge of an analogue world? If so, then my previous points apply.

Best,

Dean

Hi Ian, I'd just like to re-iterate my point about a spinning helix which travels around a hypersphere being analogous to an electric circuit. Imagine you are on the inside of a battery which is connected to a simple loop of wire which makes an electric circuit. Imagine a handle rotates clockwise from the positive terminal as seen from your internal perspective. Now trace this turning handle as it travels along the wire and arrives at the negative terminal of the battery. Which way is the handle now turning from the viewpoint of the battery's interior? Is it clockwise or is it anti-clockwise?

The thought experiment illustrates the important relationship between chirality, loops and mirror images. Incidentally, I learnt from a repeat of QI on TV last night about oranges and lemons. The aroma of a lemon is the exact mirror image of an orange and vice versa. Our olfactory sense, the first one to develop via evolution I believe, is ultra sensitive to right and left handedness of airborne molecules, which I find quite interesting.

Kind regards, Alan

Ian,

Well considered and summary look at the difference between how pure math operates versus limitations of actual measurements, at the "bird's eye view." Indeed, models and measurements are not so dovetailed as the glib idea suggests. This is particularly vexing in the quantum realm. I invite you and others to look at my essay, at http://www.fqxi.org/community/forum/topic/949. There I consider the claim that decoherence in any way resolves the measurement problem, with proposed experiments to confirm my claim that the answer is "no, it doesn't." To me, the MP transcends the issue of discreteness, because (ironically named) realist concepts per se can't fully describe our universe or deal with genuine unpredictability.

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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.

Dear Dr. Durham,

Thankyou for the reply. In my essay is a defintion of 'the Light,' which is a generalisation of the energy of a photon. Historically and conceptually this definition precede's the advent of QM.

Based upon what you said above, 'the Light' is both epistemic and ontic. Therefore the reason knowledge of 'the Light' is discontinous (at the subatomic level)is because 'the Light' itself is discontinous. Perhaps you could read my essay and let me know if you think that is correct.

Robert

Dear Ben Baten,

For an electron at rest Dx (D is Delta) is not zero, as you say, but

Dx ~ (hbar/mc),

with m being the electron mass. This is the equation (13) in the Reference 6 cited in my Essay.

Reference 6 rigorously revises these and other topics (for instance, the equation (14) gives the value of Dx for an ultra-relativistic electron with momentum p), explains why those limits Dx and Dt are not fundamental but arise only under certain approximations in the propagators (as the approximate propagators used in relativistic QFT), and corrects other claims that you and Ian are doing here.