Dear Eckard

I recommend you to read the paper of Deutsch and Lupacchini that I cited on my essay, I am sure you will enjoy it even if you will not agree with their arguments, it is a short and very nice paper. There you will find why I think classical logic is not appropriate to describe quantum reality.

Dear John Benavides,

I did not yet manage getting D. Deutsch, R. Lupacchini, Machines, logic, and quantum physics, Bulletin of Symbolic Logic 3 , September 2000. While I esteem Diana Deutsch an excellent expert, I am not sure whether the work by David Deutsch has a sound basis. I already uttered my doubt that quantum computing will ever work as promised, not for the excuses aired in a previous contest but possibly due to a fundamental mistake. What about formal logic, I abstain from layman's guesses. I merely criticize obvious to me inconsistencies in fundamentals of mathematics.

Let me give examples:

Dedekind himself admitted to have no proof for his famous cut. The devil is already in his basic assumptions.

Cantor might have got insane because he was unable to provide an already announced proof.

Kronecker understood that G. Cantor was wrong. When David Tong wrote "... everyone disagreed with Kronecker" he was unfortunately right. Kronecker's tragic was, he also intended to perform the impossible: Arithmetisierung of the continuum. Cauchy and Weierstrass preferred to build on Leibniz's pragmatism. Irrational numbers were already pretty well understood by Martin Luther's friend Stiefel. The naive elusive alephs by G. Cantor caused nothing but unnecessary trouble. Or can you give an example for any use of aleph_2? Having uttered this criticism privately, I often heard: We know, Cantor ... However. Even Ebbinghaus evades in public to call a spade a spade. Read his Lessing quote.

My position is quite clear: While real numbers must not be understood as rational numbers of very high precision as the actual infinity must not be understood as a very large number, it is nonetheless absolutely legitimate to treat them like rational ones because the difference between R and Q is only relevant if one tries to single out a particular number.

I dislike books for engineers that illustrate the function |sign(x)| v-shaped going down to zero for x=0 "for mathematical reasons". This is a benign but unfortunately not the only nonsense.

While I highly appreciate H. Weyl's honest admissions:

"We are less certain than everabout the ultimate foundations" and

"at the moment no explanation is in sight" (for PCT symmetries),

I cannot agree with his metaphors calling the rational numbers bones embedded into the sauce of reals. Well, the sauce is a good metaphor. However, the number five of rational numbers has lost its accessible identity among consequently understood reals. It is no longer a bone.

I guess, I understand well why mathematics refuses accepting logic. My suggestion the reinstate Euclid's notion of number more precisely as a limit measured from zero might hopefully provide a way out the calamities.

Did you understand my points?

Regards,

Eckard

Doug Bundy pointed me to a "clarification" by John Baez. I did not find it and asked in Physics Forum for help.

A. Neumaier replied concerning what Andy Akhmeteli and I found out independently of each other: "In principle C is not necessary even for quantum theory-"

He argued: "It might be sufficient in principle but forcing physics into the Procrustes bed of banning C would make many things very tedious - from the Fourier transform to creation and annihilation operators. How would one write the canonical commutation relation [q,p]=i hbar without using complex numbers?"

The answer can be found in my essay:

C is definitely superior but redundant. R and the cosine transform are sufficient in principle. So called "verschaffte Quantisierungsbedingung" can be written as 2 pi pq/h - 2pi qp/h = i as to understand that Plancks constant h has nothing to do with i and commutation. Both the imaginary unit and the property of non-commuting are redundant artifacts due to complex Fourier transformation from one-sided reality into the Hermitian symmetry in complex domain. Notice: Fourier transformation is based on an arbitrary omission. This and its consequences is surprising to experts of quantum theory. Nonetheless it is true.

Eckard

I reformulated my reply to A. Neumeyer as follows:

As already the title of my essay "Continuation Causes Superior but Unrealistic Ambiguity" indicates, C is excitingly superior to R which is on its part superior to R while R is once redundant, and C is twice redundant, i.e. fourfold copy of reality if we obey the undeniable property of all measurable functions of time to be restricted to what already is or at least will become past.

From this restriction follows that R and cosine transform are sufficient, in principle.

So called "verschaffte Quantisierungsbedingung" can be written as 2 pi pq/h - 2pi qp/h = i. Planck's constant h has nothing to do with i and nothing with non-commuting matrices.

Both the imaginary unit and the property to not commute are redundant artifacts due to complex Fourier transformation from one-sided reality into complex domain with Hermitian symmetry. Notice: Fourier transformation requires arbitrary analytic continuation, and it is further based on an arbitrary omission. This inevitable implies redundancy and ambiguity, which would vanish with correct return into the one-sided and real domain of reality.

Once again, who can point me to a clarification by John Baez concerning the notions number and continuum?

Hi Eckhard,

I'm sure it was on Baez's site, n-category cafe, where I was reading it. For instance, he writes about metric spaces and says: (see here)

"Let's start with a simple observation. Steve Schanuel has a paper 'What is the length of a potato?' in which he points out that a closed interval of length 1 centimetre is not very good as a ruler: if you put two of them end-to-end then you don't quite get an interval of 2cm - there's a point in the middle left over. A half-open interval would be better. Thus, the measure of the closed interval isn't really 1cm: it's 1cm+1point, or 1cm1+1cm0, or simply 1cm+1. Similarly, a closed interval of length acm has length acm+1."

I don't know if this is what I remembered now or not, but at least it's a place to start.

Dear Doug,

Thank you for the clarification that John Baez did not clarify anything but his dependence on Cantor & Co. He just added one more bewildering consequence to for instance Buridan's ass, the inability of topology to perform a symmetrical cut, the failure to correctly return to the original after cosine transform and its inverse as revealed in my Appendix A, Haseltine's problems at the origin, and Terhardt's updating of Laplace transformation.

The reason for me to deal with the matter was the inability of otherwise highly respectable mathematicians to tell me convincingly how to deal with the very nil when splitting R (blackboard bold) into R and R-. Some suggested including it into R, others into R-, others demanded treating it separately, others gave me the permission to decide at will. In short: I did not get a convincing advice.

I was forced to clarify the murky matter myself. Fortunately I grew up in the ruins of Eastern Berlin. So I can read the original papers, and the need to survive in changing political systems trained my distrust.

Let me continue later.

Regards,

Eckard

Dear Eckard,

Again, is it not a problem with defining a point? There must always be room for the concept of nothing, but how something can be located to the left of nothing makes no sense at all, unless we are talking about motion, locations and directions. I'm reminded of the "crying jag" that Lorna Sage had when she was a child, as reported by Sean Carroll several years ago, when he was discussing the doorway leading to the bewildering consequences that makes things so murky today. (See here)

W.R. Hamilton had similar misgivings about negatives and imaginaries, which he sought to eliminate by basing algebra on motion, instead of fixed forms. But as you point out, engineers can't afford to mess around with bewildering consequences. When they discard one of the side bands in SSB transmission to eliminate the redundancy of info in their transmissions, they are looking for practical results.

Nevertheless, when the redundancy is part of the symmetry of nature, as she is always balancing her act, I think it's an important objective to understand how and why she does it, even though that understanding may not have practical consequences in the short term.

When Dirac had the sudden insight that connected Heisenberg's "strange" multiplication with the familiar Poisson brackets, the physics community was trying to solve the real problem presented by real experiments. They could not, of course, see the day when the use of Lie algebras of Lie groups would run into trouble trying to extend their ideas into three dimensions.

The concepts of C and R- have worked out well for physics, and C has worked out well for engineering, in a practical sense, even if philosophers have been left bewildered and confused. For a philosopher, understanding is usually a matter of clearing out the fog, but when the fog keeps pouring out of universities and labs as it does in modern times, the hapless philosopher can only retreat to a cave and try to clear his head, as best he can.

In my cave, I contemplate the tetraktys. It starts with 20 = 1, implying that rational numbers are primordial. If so, then 2/2 = 1 must have some profound significance, and if that signicance lies in the symmetry of reciprocity, then the concept of inverses is a crucial one. As I touched on in the discussion of my essay, it is useful to think of 2/2 = 2(1-1) = 20 = 1, as 1/1, where the equality of the numerator and the denominator denote a difference of 0, as well as a quotient of 1.

When we take this ratio view, then there are definitely two real "directions" possible, with respect to 0, the negative ratio 1/2 and the positive ratio 2/1, on either side of the unit ratio, 1/1, which is equal to unity, because their is no difference between numerator and denominator. The difference, or imbalance of 1/2 can be expressed as -1, the lack of difference, or lack of imbalance of 1/1 can be expressed as 0, and the difference, or imbalance of 2/1 can be expressed as +1. So, we have -1, 0, +1, with no issue remaining as to which "side" we should assign nil to, and the fact that there is a two octave spread between them suggests that the c limit in physics may be misleading, in some cases.

Just some thoughts to share with you.

Regards,

Doug

    Dear Doug,

    Yes, yes, and yes with minor caveats. I intend continuing my other posting as promised but not yet immediately. Then I will explain why I consider the current interpretation of real numbers a chimera.

    Thank you for reminding me of even more reasons to get aware that sometimes the idea "a number is a number is a number" has unacceptable consequences. Before Terhardt wrote his updating of Laplace transformation, he faced notorious rejection of his justified criticism from all peers. Your argument is absolutely correct while the use of located and location is seemingly self-contradictory:

    "how something can be located to the left of nothing makes no sense at all, unless we are talking about motion, locations and directions".

    I would rather prefer to say: There is no negative distance in reality, and no effect precedes its cause. Non-causality is always unreal.

    I was happy that my students did not ask me how to apply the picture of a symmetrical delta impulse on a one-sided radius scale at r=0. While I hate pretending, I nonetheless felt obliged to not deviate from mandatory theory.

    What about C, my point relates to the tacit use of an arbitrarily chosen omission, of mostly either the clockwise or the anti-clockwise rotating phasor. Euler's equivalence is no problem at all. Heaviside's continuation demands the correct return to the original one-sided and real domain.

    Admittedly I do not quite understand what you meant with "cave". And of course, log(1)=0. A measure in dB is not the original one.

    I am not familiar with Hamilton. Did he continue the old fluentist ideas which were used e.g. by the Pythagoreans, Aristotle, Cavalieri, Torricelli, and Newton?

    I don't see Dirac responsible for the double redundancy in quantum theory. Already Poisson and later on all experts around Heisenberg including Kramers, Born, Pauli, and Jordan took the unrealistic bilateral time scale from minus infinity to plus infinity for granted. Accordingly, Schroedinger/Weyl used to choose a complex ansatz.

    Regards,

    Eckard

    Thank you for you essay. It is good to see people who are passionate about their field, since it brings about good discussions. I have a few remarks about the content:

    1. Page 1. Analog computing is outdated. It was immediately bound to the real behavior of lumped electric amplifiers, capacitors, and resistors. Its results could be wrong, e.g. due to ignored invariance laws.

    What kind of invariance laws are you referring to?

    2. Page 1. Some phenomena can better be described by continuous "analog" models, others by discrete models.

    There are also models which have a combination of discrete and analog aspects. Think about a clock, which has both continuous and discrete behavior: continuous behavior between the ticks and discrete behavior at the moment the tick is produced.

    3. Page 2. quantum physicists are trying to derive from created mathematics a completely discrete structure of reality.

    This is partially true. The Schrodinger equation has a continuous solution. If you talk about particle models, then it is somewhat correct. Both particles and interactions are modeled as discrete entities, often involving never directly observed virtual particles. I think there are major issues with that approach.

    4. Page 7. When Minkowski's introduced ict as fourth dimension, he confessed not to understand why it is imaginary.

    There is no real need to introduce imaginary time. It leads to more convenient mathematical expressions, but moves away from the underlying physics. See my essay for an impression that i is not needed. On my website I have a report that derived the Lorentz Transformation without ever needing i).

    5. Page 7. Planck's constant h is just required as to get a dimensionless argument. It may be called quantum of action, but it has the meaning of the smallest quantum of energy E=h-bar*omega only on condition there is a lower limit to the circular frequency. Wave guides have such a cut-off frequency for transversal waves.

    The expression and cut-off behavior is, I think, correct for the case of waveguides that you mention involving photons. However, unlike for electrons, the quantum of action of photons is equal to zero (E=pc gives 0=Edt-pdx, where c=dx/dt, for free electrons: h=Edt-pdx, more details on my web site). The expression E=h-bar*omega should really be read as omega_photon=dE/h-bar, where dE is the change in energy of an atom when the photon is emitted. Conservation of energy then says dE_atom = E_photon such that E-photon=h-bar*omega_photon, where h still pertains to the atom. E-photon=h-bar*omega_photon can be less formally written as E=h-bar*omega. So, although the quantum action of a photon is equal to zero, the expression E=h-bar*omega contains h. Lot's of details, but important for correct physical interpretation imo.

    Best regards and good luck with the contents.

      Dear Ben,

      I apologize for not yet having read your essay. Thank you very much for already responding to mine.

      1. Let me give a primitive example first. Imagine a cube with sides of length a. Volume V grows with a^3. Surface S grows with a^2: S/V=a^2/3.

      Many decades ago, in order to easily investigate the eddy currents within a small metallic disc rotating through a magnetic field of a power meter, an enlarged analog model was build. Each dimension was enlarged by the same factor. The results were horribly wrong.

      2. Yes. An similar example is to be seen in the ripples shown in my Fig.1. Notice: Those who were only familiar with the traditional FT-based spectrogram could not believe that this obviously causal and also with respect to other features more realistic figure was correct because it seemingly violated the uncertainty relation, which is to be seen valid just for discrete lines on the ridges of the continuous ripples.

      3. With "completely digital structure" I meant quantization also of space and time. Because this is hard to imagine, I asked Lawrence Crowell for his fractal picture, and he showed it. Not just for Charles S. Peirce but also for Schroedinger, space and time were continua. Hopefully you got aware that and why I consider not just point charges, line currents, singularities and the like very useful but strictly speaking unrealistic ideals, but I consider continuous functions like sin(omega t), when thought to extend from minus infinity to plus infinity, also just unrealistic fictions.

      4. I referred to what Minkowski himself wrote. By the time I will carefully read what you have to say concerning Lorentz tranformation.

      5. My emphasis was on "only on condition", meaning in principle perhaps not. Acoustical cavities and coaxial electric cables admit longitudinal waves with very low frequency, too.

      Best regards,

      Eckard

      Dear Doug,

      Andy Akhmeteli pointed me to a different "clarification" by John Baez: "Division Algebras and Quantum Theory" arXiv 1101.56904v:

      "Abstract: Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly "complex" representations), those that are self-dual thanks to a symmetric bilinear pairing (which are "real", in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are "quaternionic", in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds - real, complex and quaternionic - can be seen as Hilbert spaces of the other kinds, equipped with extra structure."

      This does of course not answer my original question, which was already answered by your hint with the "point in the middle left over". Nonetheless it shows to me that mathematicians like John Baez do not devote the due attention to what I consider important: the question of redundancy and ambiguity.

      Regards,

      Eckard

      Dear Eckard,

      While there is no negative distance in reality, there definitely is a left and right, an up and down and a backward and forward, relative to a selected position, and since space is usually defined as a set of positions satisfying the postulates of geometry, I think it important to not throw the baby out with the bathwater.

      Sir W. R. Hamilton was an English astronomer and mathematician who coined the term "vector," first explained complex numbers as coordinates on a graph (or rotation made possible by an imaginary number), and is credited with the invention of quaternions (even though a Frenchman beat him to it and understood the true nature of quaternions better).

      It was based on his ideas that Pait, arguing for Hamilton's quaternions, took on Heaviside and Gibbs, and lost the battle with their vector algebra, which has dominated ever since.

      However, quaternions found a new life in computer calculations, since it greatly simplifies the manipulations of 3D rotations and avoids gymbal lock. They were mostly revived by David Hestenes, who first recognized the value of the ideas of Grassmann and Clifford and popularized them through his modification of Grassmann's geometric algebra (GA) (see here).

      GA is slowly gaining acceptance, since it greatly simplifies vector algebra through the use of new concepts and definitions that combine scalars and vectors in something called the geometric product. It has to do with rotation and an inner and outer product.

      As I mentioned, it was Hestenes' work on GA that drew attention to the work of Hamilton, Grassmann and Clifford, and it was mostly centered on the set of Clifford algebras upon which his work sheds so much light.

      This lead me to a little known essay by Hamilton that is called "Algebra as the Science of Pure Time." Most people think that it was based on Kant's ideas of time. I don't think it has anything to do with the Greek concept of the continuum as a moving point.

      It focuses on the comparison of moments of time, where we can say that two moments may be coincident, or else one is later or earlier than the other. Of course, his development was confined to the observer's frame of reference in the abstract.

      Dear Doug,

      The paper "THEORY OF CONJUGATE FUNCTIONS,OR ALGEBRAIC COUPLES; WITH A PRELIMINARY AND ELEMENTARY ESSAY ON ALGEBRA AS THE SCIENCE OF PURE TIME" by the Irishman Hamilton does indeed play a role in the history of complex and hypercomplex numbers. Nahin deals with the certainly more important former.

      Steve Dufourny (I hope our Wallonic friend will forgive me if I misspelled his name), blamed me repeatedly for vanity. Is it really bad if I am convinced that so many so famous experts including Hamilton, John Baez, and Ken Wharton neglected a trifle? I consider essential what Claude Shannon formulated roughly as follows: While, in principle, the past is known to us but we cannot change it, the future is, in principle, unknown to us but can be influenced.

      No matter how we interpret free will, I do not see any justification for the belief in a fatalistic block universe. Is it vanity to say that tense-less mathematical physics does not yet obey what is well considered in ordinary language?

      To some extent I agree with the essay by Hadjidakis "Has the time ...".

      You wrote:

      "While there is no negative distance in reality, there definitely is a left and right, an up and down and a backward and forward, relative to a selected position, and since space is usually defined as a set of positions satisfying the postulates of geometry, I think it important to not throw the baby out with the bathwater."

      Well, we are almost free to choose the origin of our coordinate systems. Almost means: The domain of certainty ends and the domain of possibilities begins in reality at the point NOW. You certainly know the famous sentence: Give me a fix point, and I will turn the world upside down. Notions like left and right, up and down, forward and backward refer, as you correctly wrote, to a selected position. In other words, they are relative ones.

      Are there logical, natural limits? I think so. Any distance cannot be negative. Likewise, elapsed time is always positive. The sliding relative to our ordinary time scale point NOW is a fix point that allows us to turn theories upside down that are obviously leading from one paradox into the next one. Blame me for vanity. I apologize for hurting so many. I do not consider myself a PC potato watching the anticipated movie of my past and future life. Please accepts this as to grasp in what sense redundancy grows from R to R to C and so on.

      Regards,

      Eckard

      Regards,

      Eckard

      Dear Eckard,

      When we consider the approach of the pendulum to the plumb line, the plumb line is definitely in its future. If the plumb line and the arc of the pendulum's swing are divided into n and m parts, respectively, then we can describe the projected position of the pendulum on the plumb line in terms of x/n, as x --> n, and the actual position of the pendulum in terms of y/m, as y --> m.

      I think that we can all agree that a "now" moment arrives when x = n and y = m. However, the plumb line is still in the future of the pendulum at this point, since both n and m cannot be points of no spatial extent, but only magnitudes of some extent, ad infinitum. Clearly, we cannot say that the two are coincident, until both 1/n and 1/m --> 0.

      Yet, if we are to measure the diminishment of these two remaining units, we have to further sub-divide them into n and m parts, and the process starts over, ad infinitum. Hence, we see that Zeno's paradox is in full force, in this case, even though we know that x --> 0 and y --> 0, eventually.

      How do we resolve this paradox? I submit that one way is to transform the remaining y unit into its inverse. Since x is wholly dependent on y, it can serve as a measure of y's transformation into its inverse. When y's transformation is complete, x = 0.

      But what is the inverse of y, if not -y? and doesn't x --> 1, in the other "direction," as y --> -y? How, then, does x --> 0 and x --> 1, simultaneously? The only way to resolve this new paradox, as far as I can tell, is to admit that x too has an inverse, namely -x. Consequently, as y --> 0, -y --> 1, simultaneously, and as x --> 0, then -x --> 1, simultaneously.

      Such a transformation, in both cases, involves an instant change in "direction," at all "points" along the length of the units. It is only at this boundary between the two, opposite, "directions" that we can consistently define a point of no spatial extent and an instant of no temporal duration.

      The physicists describe just such a constant transformation, as the potential energy of position is transformed into its inverse, the kinetic energy of motion and vice-versa. Therefore, this approach not only seems reasonable to me, but exhibits a hint of that unreasonable effectiveness of math in the natural sciences, that Wigner points out.

      Warm regards,

      Doug

      Dear Doug,

      You intended to demonstrate that negative quantities are necessary. Well, this is true for mathematical physics with emphasis on mathematical in a derogative sense. I see so called harmonic oscillator as well as your pendulum as examples for strictly speaking unrealistic models. In reality, no oscillator oscillates for good.

      You might argue that for instance sound pressure is the alternating components of atmospheric pressure. Maybe you heard of mammals blooding out of their ears because they were exposed to 210 dB re 20 microPa. Yes, I refer to whales in water. Air cannot convey symmetrical waves of such SPL because there the negative half wave is limited to the atmospheric pressure.

      In your example, neither kinetic not potential energy is negative. If I recall correctly, energy typically corresponds to distance from origin in phase plane, and realistic trajectories of stable systems are strictly speaking inward directed spirals. What about your definitions of x and y, aren't they arbitrary?

      You wrote:"Such a transformation, in both cases, involves an instant change in "direction," at all "points" along the length of the units. It is only at this boundary between the two, opposite, "directions" that we can consistently define a point of no spatial extent and an instant of no temporal duration."

      Really? Isn't this at odds with Euclid's definitions of a point as having no extension in general and of a number as always denoting a measure, not a point?

      While quantities like distance and elapsed time do not have an inverse, differences between distances or between elapsed times can of course be negative. Consequently, the logarithm of e.g. a relation between two distances can also be negative.

      As long as physicists always prefer the "more general" R instead of the tailor-made R, people may continue mocking: mathematical physics is if three out of two men left a room and therefore one man has to come in as to make the room empty.

      Warm regards,

      Eckard

      Dear Eckard,

      You wrote:

      "In your example, neither kinetic not potential energy is negative. If I recall correctly, energy typically corresponds to distance from origin in phase plane, and realistic trajectories of stable systems are strictly speaking inward directed spirals. What about your definitions of x and y, aren't they arbitrary?"

      Well, of course, but we need labels. No one is arguing the ontology of negative cabbages. However, if the point of the plumb line, which the location of the pendulum crosses, is designated '0' then, according to normal convention, we can label the top of its swing to the right as positive 1, and the top of its swing to the left as negative 1.

      Even though both of these poles are positive, in the sense that the rise of their swings is in opposition to gravity, the location of the pendulum at either pole is the result of motion in two, opposed, "directions." If we plot the motion, and correlate it with potential and kinetic energy, we can see how the two types of energy are inverses:

      Position of Pendulum: -1, 0, +1

      Energy of Position: -1, 0, +1

      Energy of Momentum: 0, 1, 0

      Thus, the fact that kinetic energy is the inverse of potential energy does not require negative reality, only negative labels to describe the reality.

      Wouldn't you agree?

      Regards,

      Doug

      Dear Eckard,

      You wrote:

      "This does of course not answer my original question, which was already answered by your hint with the "point in the middle left over". Nonetheless it shows to me that mathematicians like John Baez do not devote the due attention to what I consider important: the question of redundancy and ambiguity."

      The reason this is so, I believe, is due to the fact that they have had to construct the 1D, 2D and 3D inverses by inventing real numbers and imaginary numbers. They invented real numbers via logic and symbols, and imaginary numbers via rotations.

      In a sense, by combining these ad hoc inventions, they have been able to increase the algebra of 20 (the reals, R), to 21 (the complexes, C), to 22 (the quaternions, H), and to 23 (the octonions, O), but the O algebra is non-associative. Hence, they run into problems with trying to use it in physical theory.

      Since the algebra of C only lacks the distributive property, they mostly use it, but evidently they can use the R and the H algebras, as well (the latter probably as H+H).

      But what they would really like is an R3 algebra that has all three division algebra properties intact. The fact that R- is redundant to R+ and that the arrow of time has to be added to remove ambiguity in physical interactions modeled with DEQs, is not immediately relevant to their challenge of pathological algebras that they are having to cope with, though it might still be after a solution is found.

      Regards,

      Doug

      Dear Doug,

      You wrote: "... we need labels."

      I agree that use of labels is superior. However, I maintain: Redundancy and arbitrariness can be avoided in principle by choosing an appropriate point of reference.

      You are quite right: "if the point of the plumb line, which the location of the pendulum crosses, is designated '0' then, according to normal convention, we can label the top of its swing to the right as positive 1, and the top of its swing to the left as negative 1."

      However, the normal convention is arbitrary.

      Just add what you wrote:

      "Energy of Position: -1, 0, +1

      +Energy of Momentum: 0, 1, 0"

      The sum is strange: -1, 1, +1

      You wrote: "Thus, the fact that kinetic energy is the inverse of potential energy does not require negative reality, only negative labels to describe the reality."

      I consider the sum of kinetic and potential energy in a closed system constant:

      sin^2(x)+cos^2(x)=1.

      "Wouldn't you agree?" I would like to do so because I need supporters in this contest, and I hope it is for you just a little step to accept what I consider a key.

      Isn't there still growing confusion not just among students who have to swallow an abundance of allegedly counter-intuitive theories from Cantor's transfinite alephs up to white holes? I observed even here in the contest a widespread readiness among laymen to spontaneously welcome new suggestions ranging from a variable Planck's constant by Constantinos Ragazas to Peter Jackson's exciting ideas and the tendency among experts to favor overly formalized "serious" speculations. Not even Georgina Parry did consequently challenge idolized theories. I consider my arguments in Appendix C strong ones. I just avoided to open one more frontal attack.

      Best regards,

      Eckard