Absolutely. One of the major differences between string theory and reciprocal system theory, is that reciprocal system theory is quantified and makes predictions in terms of physically measurable quantities. I have already mentioned the atomic number and isotopic mass, which QM cannot predict. However, there is more, much more. Nevertheless, so far, what we have in the way of calculations is the work of a few men, applying the system, whereas, in contrast, the Newtonian system has been applied for generations by armies of highly trained people.

One of the areas in which QM excels is in predicting the atomic spectra with the nuclear model, but we have so far been unable to complete these calculations with the RST model. We're working on it, though, and have every expectation of success.

This guy is not a member of our Society and never has been as far as I know. He makes no attempt to communicate with us, but has been trying to promote Larson's ideas for years from his own perspective. Not that there's anything wrong with that, there isn't, but I think people's impressions are sometimes less than what they might be, if they don't get their information from members of our Society. ISUS was formed in the mid seventies, and was specifically organized to promote Larson's work. We are not perfect either, by any meams, but we are determined to present Larson's work as honestly and straightforwardly as possible, having associated with him for many years before his decease in the early nineties.

While he was alive, there was an active interest in his work as indicated by his frequent invitations to address university audiences, NASA personnel, and other interested groups. After he passed away, things died down quite a bit until after the turn of the century. Today, as should be evident from my postings in this forum, theoretical physicists are more prepared than ever to seriously consider his work.

Brilliant Excal, very fine exposition of the possible upcoming change in the fundamental background of physics, as force a function of motion, where space/time is a reciprocal function of motion. In your RST part III, you write: This can be understood, I think, in the sense that even when a reference frame is at rest, say at v=0, the motion continues to exist as "space moving" so that any material quantity at "rest" is in fact already in motion. This is supported by the first and second postulates:

"The first fundamental postulate posits the existence of motion, as the sole constituent of the universe; ...the second postulate, that posits 'motion expressed in space' presents a conundrum, similar to the one we see arising from the conflict of GR and QFT: how do we describe motion in space, when space is defined as an aspect of motion?"

One way out of this conundrum was to simply make space's "motion" (which must have some vectorial direction) as a directionless scalar. At least, this is how I understand it so far. Time as a reciprocal of motion can then be defined as dt = ds/v, which is the same as ds = v*dt; however what is unclear to me, is time now a fixed unit of measure, or can it be a variable as per SR? The choice of "time" units of measure may then affect how we see symmetries in physics, since using time as a unit of measure in space and scalar motion would change the symmetries, I should think.

This problem of symmetries derived from "time as a reciprocal of space/motion" may also have its equivalence in the "conservation law of energy". This is how I see it, and perhaps this could be understood as radiation electromagnetic energy conserved within parameters of its scalar motion of space. In effect, if the Energy Conservation Law is incomplete, meaning it is missing some component to make it whole and totally symmetrical, then there may be a reciprocal form of energy that is (perhaps) gravitational in nature, which is also part of the conserved total. This would mean total Energy is both as an electromagnetic wave and something in gravity (of which we are still ignorant) to give us the fully conserved value. We know both are responsible for motion, both act as a force, and yet in today's fundamental background of physics they are essentially divorced from each other. At present, this is the reason we had not reconciled the two to date. Yet, if Larson's system could show that motion resulting from gravitational force is related to scalar "motion" of moving space, then we should be one step closer towards a system where radiant energy and gravitational energy are two aspects of the same motion. Is this sensible, or am I simply saying it all wrong?

Thanks for bringing up Larson's alternative Reciprocal System of Physical Theory, Excal. I just hope I didn't botch it up (am assuming I did!) too much, trying to explain what I may not yet really understand. I'll read the math next... Still reading...

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Well, the important thing to try to grasp is that space and time don't have any existence apart from motion. What often happens when we first start thinking this new way is that we try to visualize space as something that exists apart from motion in some way. However, just as the left and right ends of a rod can be described as two reciprocal aspects of the rod, which couldn't exist without them, and because we could describe them independently in terms of their different positions, the direction they face, etc, and could treat them in the abstract as if they actually had an existence in their own right, we might begin to forget that they are only aspects of the rod, that the rod is the only thing that actually exists, we would eventually run into trouble.

So it is with scalar motion. Motion is what actually exists. Space and time are only its two reciprocal aspects. No matter how hard you try, you can never have one without the other, just as you can never have a left end of a rod without a right end.


Actually, to be clear, time is an aspect of motion that is the reciprocal of the other aspect of motion, space. In SR, the measure of space and time change to acommodate the speed of light, which must remain constant relative to any inertial frame of reference, but, in the RST, the measures of space and time are discrete values that are absolute, and the constant speed of light in relatively moving inertial frames is accounted for by motion in time, dt/ds, the effect of which is noticable only at relativistic speeds. Therefore, there is no need for adjusting the length of measuring rods, and the rate of clocks to maintain the constancy of the speed of light. See K.V.K Nehru's paper on this entitled "Precession of the Planetary Perihelia Due to Co-ordinate Time," at http://www.reciprocalsystem.com/rs/c...planetperi.htm

I think I understand what you are getting at, and it's an astute observation. However, the details differ somewhat, but that's to be expected. In the RST, the second law of thermodynamics becomes a relatively minor player, since the major cycle of nature is a process whereby matter is transformed over time through the action of gravity. It cycles, over billions of years, between two reciprocal sectors of the universe, which Larson named the material and cosmic sectors. In the material sector, in which we live, the equation of motion is defined on the "low speed" side of unity where ds/dt = 1/(1+n). However, in the cosmic sector of the universe, motion is defined on the flip side of unity, the "high speed" side, where dt/ds = 1/(1+n) as well, but from our perspective it is ds/dt = (1+n)/1. Thus, the perfect symmetry of space/time, provides two sectors of motion relative to the unit datum, or natural reference system of uinversal progression. Gravity is the driver of the evolutionary process in one sector, while inverse gravity is the driver of the process in the reciprocal sector. Thus, as you suspect, total energy is conserved in the symmetry of the system.

However, in the LST, because the definition of motion is limited to an object's change of locations over time, ds/dt, the magnitude of its velocity is limited to 1/1, or c. Nevertheless, dt/ds, is also motion in the RST, therefore matter can move at speeds higher than c, but the motion is not space motion, it is time motion. This gives rise to bizarre astronomical phenomena from the "low speed" perspective, because it's not recognized for what it is.

No problem. In the next post, part VI, we'll be looking at the contrast between the application of mathematics in the LST, based on PDEs, and the application of mathematics in the RST, based on the rational number system as a mathematical group. I think you will find it most interesting.

Regards,

Doug

In the Newtonian system of physical theory, which I'm going to presumptiously refer to as the legacy system of physical theory (LST), for brevity's sake, everything is based on the function x(t), given Newton's laws of vectorial motion, while in the Reciprocal System of Physical Theory (RST), everything is based on the function x(t), and a new function, t(x), given the new interpretation of the nature of space/time, as the two reciprocal aspects of scalar motion.

In the LST, the focus on forces to describe and explain the relationships between the properties of mass, magnetic moment, and electrical charge, necessarily involves understanding the coordinate path of an entity, because changes in the coordinate path, or location, of a physical entity constitutes the condition which gives rise to force, given Newtons laws. However, the science of dimensional analysis, initially employed in the investigation of forces, quickly becomes tedious and unwieldy in this respect.

Fortunately, however, it was discovered that such studies could be simplified to a great extent by expressing these relationships in terms of energy, instead of force. Thus, the total energy of a system, in terms of its kinetic and potential energy, can be formulated as Legrangian equations of motion, leading to Hamiltonian mechanics, taking advantage of the principle of least action to simplify the study of forces.

Consequently, the Legrangian and Hamiltonian fomalisms of mechanics have led the way to understanding how the properties of physical entities are related, including easing the transition from the concept of force associated with a coordinate path, an x(t) function, to the concept of force associated with a coordinate field, written as a q(x) function, where x represents both space and time coordinates. This mathematical reinterpretation of the x(t) function, enabled the reformulation of Newton's universal law of gravity into a function of a field, the G field, the gravitational field, and eventually, it enabled Maxwell's formulation of the magnetic and electric equations of motion as fields as well, the E and B fields, and their unification in the electromagnetic field theory.

The dramatic success of these and subsequent developments have led to unimaginable advances in engineering and technology that have harnessed this knowledge for the material benefit of mankind, and led to the enshrining of theoretical physics as the foundation of modern civilization and enlightenment. Interestingly enough, it has also led to the conviction among physicists that the "unreasonable effectiveness of mathematics in physics" is the hallmark of modern theoretical physics.

However, it should be clear that removing the requirement that we need a background of space and time coordinates in order to define motion, and moving up to a more fundamental understanding of the nature of space and time, as the reciprocal aspects of a universal motion, existing in three dimensions, and in discrete units, is going to have an enormous impact on the need for these complex mathematical formulations, used so effectively in the LST research program, namely the ubiquitous use of partial differential equations.

Moreover, since the primary aim of the RST program is not to determine how the given properties of physical entities relate to one another, as it is in the LST program, but rather to determine how the observed properties of physical entities originate from more fundamental entities, namely the discrete units of space/time, the mathematics of the RST most likely will be dramatically simplified, at least initially, but, by the same token also, it will likely exhibit another surprising and enlightening aspect of the "unreasonable effectiveness of mathematics in [theoretical] physics."

In the RST definition of motion, the postulate of the reciprocal relation of space and time, and discrete units of motion, constitute a major move of physics into the rational number system that is closed under non-zero division. Since the progression ratio of space and time can never be zero (time and space never stop progressing), this means that the system itself is indeed closed, containing its own boundary. More importantly, this means that the mathematics of the RST is a group, the ultimate closure structure.

The significance of this is that the Reciprocal System mathematics can be encapsulated in one sentence: the RST mathematics is the set of all properties that can be conserved by the rational number system group. So, just as special relativity is the study of all properties conserved under the Lorentz group of transformations, the RST is the study of all properties conserved under the rational number system group of transformations. The basic concept of group is symmetry, and, as mentioned in previous posts in this series, Emmy Noether proved that symmetry is the conservation law of nature. This is why mathematical groups have such predictive power and have assumed such a prominent role in modern theoretical physics, starting with Murray Gell-Mann's prediction of the omega minus particle on the basis of completing the symmetry of a group. [6]

With the progression ratio ds/dt defining the mathematics of RST motion, the group structure implies that the inverse ratio, dt/ds, is also closed. Thus, the symmetry of the RST is the perfect symmetry of the group, which is ds/dt = dt/ds = 1/1. With the possibility of zero eliminated, the datum of the system becomes this operational definition of unit magnitude, and any displacement of the progression rate of space or time, relative to its reciprocal aspect, "breaks" the fundamental symmetry of the unit ratio, but conserves it in the symmetry of the group. In other words, |1/n| is equal to |n/1|, because there is no way to distinguish the two magnitudes from the point of view of the datum of the system, 1/1.

This is interesting, because it sheds new light on the meaning of the natural and integer number systems as well as the meaning of Euclidean geometry, which is assumed to be a property of the universe of motion, in the second fundamental postulate of the system. The difference between the natural number system and the integer number system is found in the negative numbers of the integer system that don't exist in the natural number system. Both the natural number system and the integer number system start with the number zero, an enigmatic concept to say the least, but one we have come to take for granted.

However, because the negative numbers are the mirror image of the positive numbers in the integer system, zero stands between -1 and 1 in that system. Yet, while zero quantity makes some sense in the natural number system, it is completely out of place in the integer system, since, strictly speaking, zero is not actually a number and therefore cannot be an integer. The Merriam-Webster dictionary definition of the word integer reveals the subtlty involved. It defines an integer as, first, "any of the natural numbers, the negatives of these numbers, or zero," which explicity excludes zero as a natural number, and, secondly, as "a complete entity," something, by definition, zero is not.

Hence, something is obviously askew in our need for the quasi-number zero, but it's difficult to define exactly what it is. Obviously, we can't manage without it, yet we can't strictly define it as a number either. As it turns out, in the RST definition of motion, the quasi-number zero doesn't exist. In listing the integers of the integer system defined by the operational definition of an integer, zero disappears. This is because, operationally, the definition of the zero datum, the system's point of reference, is 1 = 1/1; that is, the difference between negative 1 and positive 1 is the operationally defined difference between 1/2 and 2/1, and the operational magnitude that stands between these two magnitudes is 1/1, not zero. Hence the integer number system is the set of integers operationally defined by the ratio of natural numbers, which excludes zero, because it is not a natural number. It is true that it is included in the mainstream list of integers, but including it in the list of integers is misleading, if the difference between an operationally defined magnitude and a quantitatively defined magnitude is not understood.

(See continuation in following post)

There is another important aspect to this number system as well. Not only can integer magnitudes be interpreted operationally, or quantitatively, but also the difference can go the other way, wherein the ratio of two integers may be understood operationally or quantitatively as well. Thus, in the quantitative interpretation, the ratio of 1/2 = .5 is not a magnitude equal to 2/1 = 2. However, in the operational interpretation, the magnitude of the ratio 1/2 = 1 is a magnitude equal to the ratio 2/1 = 1, just as the unequal magnitude on one side of a pan balance is no different than the same imbalance, if it were to be placed on the other side of the balance. The only difference is which side of the balance the magnitude happens to be found. We can distinguish this in the number system by the use of the "direction" concept of positive and negative, and so we do in the integer number system, but there is no difference in the magnitudes themselves. Hence, it makes no sense to write operationally defined magnitudes as ratios of signed integers. Operationally defined integers are all simply magnitudes. The "direction" we use to distinguish them is only an indication of which "direction" in the ratio, relative to the unit ratio, the magnitude lies. We can say one is "above" unity, while the other is "below" unity, or we can say that one is "positive" and the other "negative," but relative to the datum of the system, 1/1, there is no material difference in the magnitudes themselves.

One method that has been found to encapsulate the nature of this relationship of the integer number system to the rational number system is an adaptation of a cellular automata (CA) rule, designated rule 254, by Steven Wolfram. [7] This automaton is, to Wolfram, one of the most uninteresting of all automata generated by CA rules: a perfectly symmetrical triangle as shown in figure 1. However, by slightly modifying the meaning of the output of rule 254, we obtain an operational definition of integer magnitudes; that is, if we define the center column of rule 254, as the demarcation between the reciprocal integers, the operation indicator served by the '/' symbol in mathematical notation of rational numbers, and then define the total number of units to the left of the center column as one integer and the total number of units to the right of center as the corresponding reciprocal integer, then as the rule proceeds and adds one unit on each side of center to the accumulated total number of units on either side, as each successive row is generated by the rule, the result is a visual graph of the 1/1 reference datum of the rational number system expressed as a progression of reciprocal integers, the ratio of which is constant at 1/1.



Figure 1. Wolfram's CA Rule 254.

Thus, we start with row 1, which contains one black cell, then row 2 has two black cells on either side of the operation symbol, or 1/1. When row 3 appears, there are 2 black cells on either side, or 2/2, which is also a 1/1 ratio, and so on, ad infinitum. Therefore, the output of CA rule 254 is the integer 1, operationally defined as 1 = 1/1, which is actually the reference datum of the rational number system, sometimes mistakenly denoted as zero, as explained above.

To generate the integer commonly designated 1, or -1, which is really 1 = 1/2, or 1 = 2/1, as also explained above, we need to change the rule. Now, I don't know if there is a CA rule that will do this, although there may be one, but, nevertheless, I do know there is an algorithm, that I have dubbed the progression algorithm (PA), that will reproduce the pattern of CA rule 254 and also a deviation from this pattern, which will produce the 1/2 ratio, or the 2/1 ratio.

The PA adopts the operationally interpreted rational output of CA rule 254, and designates the left side integer as the space progression, and the reciprocal, or right side integer, as the time progression. Thus, the PA corresponding to CA rule 254 is a representation of the RST's natural reference system as defined by Larson, as shown in figure 2.



Figure 2. The Natural Reference System PA.

Obviously, we can use this to study the properties of the system. For instance, the sides of the triangle represent the unit ratio boundary, since any change in the ratio from 1/1, would immediately change the location of the sides, breaking the symmetry of the system in the process. Also, we can visually see the meaning of "direction" in our system, as the left unit boundary is distinct from the right unit boundary relative to the center column, but at the same time they are one and the same boundary, viewed from different points of view, giving us at once a clear understanding of the meaning of the concept of reciprocity, as the symmetrical growth of one entity with two reciprocal aspects.

We also clearly see the concept of discrete units of space and time in the PA, but only as the reciprocal aspects of the units of progression, or the rows. Without the reciprocal integers, we would lose the definition of the row, without the row, we don't have a progression. Therefore, one cannot exist without the other. There are other properties of the PA, which give us great insight into the RST, but we will have to wait to consider these until we can take up the subject again in the next post.

I hope that I have been able to show in this post, that, while the mathematics of the RST are based on the deceptively simple set of rational numbers, there is a lot here that we are not used to perceiving in connection with this number system, and also, that the fundamental symmetry of the system portends great things to come, for as some great philosophers have noted in the past, many times, out of that which is small, proceedeth that which is great.

References:

6) Jon Hays, "Arithmetic Redux", http://members.fortunecity.com/jonhays/redux.htm

7) Steven Wolfram, Mathworld at Wolfram Research, http://mathworld.wolfram.com/Element...Automaton.html

Are you talking about predicting the mass defect, or the most stable isotope?
Is there a page that would show me how to do the calculation that I mentioned earlier (how long it would take for a mass to fall a certain distance in a 1g gravitational field.)? As I said, I'm may seem a bit impatient, but it's the maths that would give me the warm fuzzy feeling about the approach.

I hope you don't get too impatient with me, Fortis, but you have to understand that once you have the mass, magnetic quantity or electric charge, the standard LST equations apply in calculating the interaction of that mass, magnetic quantity, or charge with another. The thing that the RST provides is the calculation of the mass, magnetic quantity, or charge, given only the unit of space (4.558816 x 10^-6 cm), and the unit of time (1.520655 x 10^-16) of the system. This is something that no LST theory can do, so it's big news.

However, in order to carry out the calculations of mass, say, of the electron, proton, neutron, or the atom of a given element, or the magnetic moment or charge of an electron, proton, etc, you first have to learn how to do it. To learn to do it, you have to understand the concepts I'm introducing in my posts, because it's a whole new ball game, if you know what I mean.

Now, I understand that the explanations are a little verbose, but I find it necessary to motivate the conclusions involved, as well as expound them, so this is why I'm taking it so painstakingly slow. However, we will get to the calculation part in due course, if you can stick with me.

As far as the mass defect goes, in the RST model of the atom, protons and neutrons don't exist as such within a "nucleus" surrounded by electrons. So the missing mass, called mass defect, cannot be attributed to the "binding" energy between these particles, as it is in the LST model. In the RST model, it is due to secondary mass effects, which will be explained eventually. The average isotopic mass calculations for a given element have to do with an isotopic build up process involving a magnetic "charge" and interactions with neutrinos, so it's quite involved. The only way to understand it, is to understand how the model is constructed from discrete units of motion.

It looks like a "double helix" to me, where the positive electron values are one strand, and negative ones are the other. Otherwise, it is a more intuitive illustration than the standard box format.

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somehow I am getting the suspition that in the end all physical calculations are going to be the same, except that the philosophical background is going to be different. But we will never be able to observe this background.
Just my 2 eurocents at this point

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善數, 不用籌策 (shàn shù, bù yòng chóu cè)
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“A good scientist has freed himself of concepts and keeps his mind open to what is”
道德經, 二十七 (dào dé jīng, 27)

That's an interesting observation!

The assumption of the Newtonian system is that matter consists of a composite of particles. The fundamental particles form atoms and atoms form matter. The aim of the system is to classify these constituent particles according to how they interact. This means finding the way to calculate (predict) these interactions given the properties of the particles. Thus, the standard model, which represents the "final answer," the best answer we can come up with so far, identifies three families of particles classified according to a set of interactions that we have been able to identify.

However, the model must be given the properties of these particles before these calculations can be carried out. I think it's something like 19 to 24 parameters that must be obtained from observation, before the model can be constructed. This is not what we want for a final answer. We want to be able to explain the properties of radiation, matter, and energy in terms of fundamental entities, but it appears that the Newtonian research program isn't going to get us there. We are stuck. We are stuck with the standard model of matter, and we are stuck with a separate model of gravity. It's an awful, and unsatisfactory (ugly and adhoc in the words of Hawking) result of the program.

Now, having said that, we also have to recognize that the Newtonian program was never intended to do what we are now demanding of it; that is, its aim was to classify a few particles according to a few interactions, and, for the most part, it has done that, and it has done it extremely well. But now, we want to go beyond that, and the program simply is not up to it. We need a new research program to go beyond the standard model.

I'm suggesting that the Reciprocal System is just the research program to do what we want. So, we are not talking about a specific theory here, but a new research program that will enable us to construct physical theory on a completely different basis. One that will not need observed properties, and a myriad of "constants," to describe and explain all the physical properties of all physical entities. The system to do this is based on a new type of motion, a motion that is posited to be the intrinsic motion of matter, never understood before.

However, the thing I must try to make clear is that, if this is true, the dynamics of the new, intrinsic, motion of matter, does not replace the motion of matter, nor the principles that we have learned, and the methods we have devised that enable us to describe how the properties of matter relate to one another. In other words, all that has been accomplished by the Newtonian system is not invalidated and replaced by the Reciprocal System, it is simply expanded into a new realm that only a new interpretation of the nature of space and time could achieve.

But please understand, though philosophy necessarily must be a big part of the description of the new research program, the theory developed to date under the new system has achieved a great deal. It has calculated properties of matter, it has eliminated "constants" of nature, it has resolved many conflicts of current models of electrical and magnetic phenomena, it has predicted major astronomical events before they have been observed, it has unified the known forces into one primary force equation, and, much, much more. It's not just philosophy.

However, what it has not done is provide a new way to calculate the interactions of matter. We already know how to do that.

Would it not be simpler to say that n/1 integer times its inverse 1/n is always equal to 1 = 1/1 ? In this way, the (non) integer zero makes sense: as n increases to infinity, 1/n decreases to zero. So what you get is n/1 * 1/n = 1, of necessity always, and zero is retained. Therefore, if it can be used this way, ds/dt = 1/1 = 1, so that the reciprocallity of space/time symmetry is conserved. Would this explanation alter the balance of what "integer" based RST math represents? Taking it to any integer, say 3, then 3 * 1/3 = 1, etc., just reciprocal values of ds and dt, but always within the symmetry of unity. At n = ds = 'space to infinity', then dt = 1/n = 'time to zero', for example, yet symmetrical.

This would also work, I think, for x^2 + y^2 + z^2 = c^2*t^2, as per your earlier illustration: x^2 + y^2 + z^2 - c^2*t^2 = 0. The conclusion drawn would be that at t=0, space is contained on a point in motion. Would this be right, or am I off on an imaginary tangent here?

Is this definition of unit magnitude within the parameters of math you are presenting?

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This is a good point. However, defining zero as an integer = 1/infinity just shows the inverse compliance of the group, it doesn't enable us to place zero between the negative and positive integers in the integer number system, as if it were the datum of the system. We start with the natural numbers, which does not include zero. Then we operationally define the integer number sytem from the natural number system, which generates the positive and negative integers, as described. These extend from 1 to infinity, because the natural numbers extend from one to infinity, but, in a sense as you point out, the inverse of infinity is zero, so it could be understod as infinity/1 * 1/infinity = infinity/0 at the far end of range. Fortunately, however,the discrete unit postulate rescues us from the logical impasse this presents, as I hope to show eventually.

I wrote

That was blatant error, as obviously infinity/1 * 1/infinity does not equal infinity/0. I was too tired when I was writing the above post. Obviously, the above equation would actually equal infinity * 0 = infinity, not infinity/0. However, infinity times its inverse, 1/infinity, is actually equal to infinity/infinity, which is equal to 1/1. We must remember that 1/infinity is NOT actually, zero, although it's conveniently regarded as such many times. In fact, given Zeno's paradox, zero is not any more attainable than infinity, so the most acurrate position to take operationally is that infinity times inverse infinity = 1/1, or 1, as you say. Whether it means that "at t=0, space is contained on a point in motion," I don't know.

How does this handle time invariance, i.e. t->t+Delta_t?

In conventional physics, if the physics is unchanged under the translation
t->t + Delta_t
then this usually indicates that energy is conserved. What you say above seems to indicate that there is a special t=0?

[edited to expand a bit on what I was saying...]

I'm not sure what "this" refers to. I was quoting you, and I don't now what the statement in the quote means.

If we arbitrarily pick a point t0 in the progression, and then pick another point at t1, one unit later, then t1 - t0 is equal to 1.520655 x 10^-16 seconds. The system makes no assumptions as to the beginning of the progression.

The same law of energy conservation would hold, if nothing changes but time. However, this applies to particles of matter, as in conventional physics, so the intrinsic motion, the motion that consitutes the particle in the RST is not changed unless the particle changes locations (its physics changes), just as in conventional physics.

As we have seen, in the RST, the initial state of the universal progression, the space/time unit ratio of scalar motion, is 1/1, or an increase of one unit of space for every unit of time increase. The result of this condition is nothing; that is, without some deviation from unity, the perfect symmetry of the initial state is a perfect void, or, we might say, nothing is perfect. However, the first fundamental postulate of the system posits the existence of discrete units of motion, existing in three dimensions. Therefore, we must assume that this perfect symmetry is "spontaneously" broken. The idea of spontaneous symmetry breaking (SSB) is a cornerstone of modern theoretical physics. It occurs in a situation, where, given a symmetry in the equations of motion, solutions exist that are not symmetric. Thus, a non-symmetrical solution to the universal progression equation, represented by the unit ratio PA, is any ratio wherein the magnitude in the denominator or the numerator of the ratio is greater than one. In other words, the symmetry of the initial state is broken when the space/time ratio of the motion is something other than 1/1, such as when it is 1/n, or n/1, where n>1.

Since in the universe of motion, the fundamental entities are units of motion, we cannot add units of space or units of time to the progression ratio directly. Therefore, any deviation from the symmetry of the unit ratio must occur as a result of the dynamics of the motion itself. Fortunately, such an alteration in the condition of the progression can occur, if the "direction" of the progression reverses; that is, if the continuous pattern of increase in the number of units in the space or time aspect of the progression is changed, from a pattern of continuous increase to an alternating pattern of increase/decrease, with respect to the reciprocal aspect of the progression, then a displacement in the magnitude of the reciprocal aspect of the progression will occur, breaking the symmetry of the unit ratio. The result will be a change from the 1/1 progression ratio to a 1/2, or 2/1, progression ratio, depending upon which aspect of the progression changes its "direction" pattern.

In Figure 3, a "direction" reversal occuring in the space aspect of the progression produces a 1/2 space/time ratio in the PA. Notice that the "direction" of the arrows in the space cells indicate a continual "direction" reversal in the space progression, thereby constraining the increase in the space aspect of the progression (the left side of the diagram) relative to the time aspect, which increases normally as the progression proceeds downward. Hence, the space aspect alternately increases and decreases, confining it to one unit of space, and decreasing the number of times it increases (the arrow points to the left) by half. Therefore, after n units of proression, the number of units of time increase is always n, but the number of units of space increase is 1/2 n, producing a space/time ratio of 5/10 after 10 units of progression, a ratio of 10/20 after 20 units and so on, ad infinitum.



Figure 3. 1/2 Time Displacement PA

Alternatively, if the "direction" reversals occur in the time aspect of the progression, the displacement occurs in the reciprocal, or space, aspect, producing a space/time ratio of 2/1, as shown in figure 4.



Figure 4. 2/1 Space Displacement PA



Clearly, the progression of the reversing aspect still exists in this new state, it's just that it is now confined to one unit of space, or one unit of time by the continuous "direction" reversals. This type of motion differs from the uniformly increasing motion of the 1/1 ratio shown in the PA of figure 2, which is simply a translational scalar motion, in that the "direction" reversals constitute a scalar oscillation of the space aspect of the motion, as opposed to the former uniform translational motion of the unit ratio. Also, it is clear that the magnitude of the new motion is one unit, or c, given the size of the natural units of space and time that we derived from the Rydberg frequency previously, since 1/2, or 2/1, interpreted operationally, are both equal to 1, although in different "directions." Larson designates the 1/2 progression ratio as the negative magnitude, and the 2/1 progression ratio as the positive magnitude. In addition, it is helpful, in refering to the two magnitudes, to call the 1/2 ratio "space motion," and the 2/1 ratio "time motion," although this concept of time motion is nothing like the time motion of science fiction, where it indicates time travel, as if one could travel back along the time progression to a previous point. Time motion in the sense we will use it simply means motion that has space/time dimensions that are the inverse of the dimensions of velocity, or t/s.

With the SSB produced by the "direction" reversals in the progression of one or the other of the reciprocal aspects of motion, we see that unit motion is asymmetrical; that is, a positive or negative magnitude of scalar motion, is one unit removed, in one "direction" or another, from the reference datum of unit progression in accordance with the integer number system. Obviously, all magnitudes greater than one in either "direction" of the integer line represent magnitudes comprised of multiples of these fundamental magnitudes. However, this mathematical characterization of the two states of scalar motion also needs a physical interpretation. This can be accomplished using a "worldline" chart commonly used to illustrate the spacetime concepts of relativity theory. In these 2D charts, the 3D spatial coordinate position of a physical event is plotted on one axis, while the 1D "position" of time is plotted on a perpendicular axis. In our adaptation of this concept, we plot the scalar expansion of the space aspect of the progression on one axis, and the scalar expansion of the time aspect on the other. Hence, the reference datum of the 1/1 space/time ratio is plotted as the 45 degree diagonal between them, and represents c speed. With space progression plotted horizontally along the x axis, and time progression plotted vertically along the y axis, as shown in figure 5, speeds below unit speed, c, are in the area above the diagonal, while speeds above c speed are located in the area below the diagonal.

Of course, speeds above c speed are not recognized in the LST, but remember, in the RST, these speeds are not space motion, s/t, but are time motion, t/s. Therefore, an RST entity, such as the 1/2 magnitude, with space/time dimensions s/t = 1/2, would be plotted above the unit line, while the inverse entity, with space/time dimensions t/s = 1/2, would be plotted below the unit line. Since, the space motion entity, s/t = 1/2, is confined to one space unit, while time progresses normally, it only increases in the increasing time "direction," as indicated by the vertical red arrow next to the circled S in the diagram. Conversely, the time motion entity, s/t = 2/1 (t/s = 1/2), is confined to one unit of time, while space progresses normally, therefore it only increases in the increasing space "direction," as indicated by the horizontal blue arrow next to the circled T in the diagram.



Figure 5. The RST Worldline Chart

In summary, then, on the strength of the SSB principle, the perfect symmetry of the uniform progression is "spontaneously" broken two ways by "direction" reversals that occur in the uniform progression patterns of either the space aspect or the time aspect of the space/time progression. When the reversals occur at some point in the space aspect of the unit ratio, a unit magnitude, 1/2 = 1, of space motion results that is composed of a scalar motion oscillating within one unit of space over time, ds/dt. Conversely, when the reversals occur at some point in the time aspect of the unit ratio, a unit magnitude, 2/1 = 1, of time motion results that is composed of a scalar motion oscillating within one unit of time over space, dt/ds. The mathematical relationships of these states is shown in the PA diagrams of figures 2-4, while the physical relationships are illustrated in the worldline chart of figure 5.

Eventually, we will explore how these units can be combined to form theoretical entities that we can compare to observed physical entities, and how their properties may be calculated. We will also see how the arbitrary units of the LST can be converted into the natural units thus derived in the RST, and how doing so explains the constants of nature and eliminates them, when the physical dimensions of various properties and relationships between them are expressed in natural units of space and time.

Ok, now we have seen how that the perfect symmetry of the Reciprocal System's initial state, the space/time ratio equal to a 1/1 progression, is the natural reference system of motion, the physical "zero" of the system, from which all physical activity is measured in the RST, and we used the PA in figure 2 above to generate this state of scalar motion as the unit progression ratio of space/time. Then we've seen how the symmetry of the initial state is spontaneously broken by "direction" reversals in the uniform progression of the space or time aspects of the motion, producing a second state of the system that deviates from the natural reference system by one unit in one of two possible "directions," one of which is designated "negative" and the other "positive." We used the PAs in figures 3 and 4 to generate these two progression ratios. We've also seen how generating these two possibilities of unit magnitude on either side of the unit ratio corresponds exactly to the initial three numbers of the integer number system, generated by the operational interpretation of magnitude that the rational number system generates. In fact, we can easily summarize all this information with just a few numbers:

1/2, 1/1, 2/1 = -1, 0, 1

Of course, mathematically speaking, any number in the integer number system can be obtained as a combination of these three numbers, from the number 2 to infinity in both "directions." However, physically speaking, this would be quite cumbersome for very large numbers, and, intuitively, one would think, quite limited, and, actually, Larson realized that there is another possibility, another state of the system, a third state, although he didn't use the terminology I'm using, and he didn't know about the PAs, or use a worldline chart to explain these things. However, it turns out that the third state results when the two possibilities of the second state are combined; that is, a new progression ratio is generated by combining the progression ratios in the PAs in figures 3 and 4. This third state breaks the symmetry of the second state in a way that alters the pattern of the "direction" reversals, changing it from a continuous pattern of reversals to a periodic pattern of reversals, as shown in figure 6.



Figure 6. Periodic Reversal Progression Ratio

For the moment, we will skip the details of how the third state results from the combination of the two possible second states. However, we can see from the PA in figure 6 that the pattern of the progression ratio now consists of a series of two "direction" reversing units, repeated over and over again, separated by a non-reversing unit; that is, in this instance, the "direction" of the space progression reverses in progression unit 2 of row two, from increasing to decreasing, in unit 3 of row three, from decreasing to increasing, but, then, in unit 4 of row four, no "direction" reversal occurs. In unit 5 of row five, the "direction" reversals commence again, and the same pattern is then repeated, ad infinitum. The resulting space/time progression ratio of this pattern is s/t = 2/3, since, for every three units of time progression, there are two units of space progression, as can be verified in the PA of figure 6, by counting the two left pointing arrows accumulated on the space side that correspond to the accumulation of three right pointing arrows on the time side of the diagram at unit 4 of row four of the progression. This pattern is repeated every three units of progression from row four on, so that at row seven, the total accumulated space/time ratio is 4/6, at row 10, it is 6/9, and so on, maintaining a constant space/time progression ratio of s/t = 2/3.

Notice, however, that the operationally defined magnitude of the space/time progression ratio 2/3 is equal to 1, as is its inverse 3/2. Thus, the third state, a combination of the two possible forms of the second state, also has two "directions" of magnitude, one negative, the time displaced unit, and one positive, the space displaced unit, as shown in figures 7 and 8.



Figure 7. Unit Periodic Time Displacement



Figure 8. Unit Periodic Space Displacement

However, we can see that mathematically, when we combine the second state progression ratios, 1/2 and 2/1, we get

1/2 + 2/1 = 2/2 = 1/1,

but, even though mathematically the magnitude of the ratio of 2/2 is the same as the ratio of 1/1, namely 1, we know from the worldline chart of figure 5, that, physically, combining these two unit progression ratios is more than a mathematical operation; that is, it's a physical operation as well. Indeed, it's a combination of the S and T units of figure 5, and we know that the S unit is confined in space, but progressing in time, while the T unit is confined in time, but progressing in space. Therefore, the space/time relationship between them is unit space/time, or c speed. Hence, the possibility exists that the progression of space could bring a T unit into contact with an S unit, while the progression of time could bring an S unit into contact with a T unit. Significantly, however, in combination these units form an ST unit that retains the characteristics of its constituent components; that is, the oscillating unit of space and the oscillating unit of time form an oscillating unit of space/time that progresses in both space and time, the same general characteristics of the physically observed photon. We can depict this combination on our worldline chart as shown in figure 9.



Figure 9. Physical Combination of S and T Units

Interestingly, the magnitude of the initial ST unit can be increased in both "directions," by adding S or T units indefinitly; that is, the 2/3 = 1 ST unit, plus an S unit, 1/2, equals a 3/5 = -2 ST unit, while the 3/2 = 1 ST unit, plus a T unit, 2/1, equals a 5/3 = 2 ST unit. Hence, once again, we see the integer number system emerging from the rational number system, only now the rational number, the ST unit of the periodic PA, is a complex number in the sense that it contains three ratios simultaneously - the 2/3 and 3/2 ratio and the 2/2 ratio. Hence, mathematically, the initial ST unit forms the integer sequence

2/3, 2/2, 3/2 = -1, 0 , 1,

which can be extended in both "directions" infinitely, by the addition of S and T units to the combination.

...3/5, 2/3, 2/2, 3/2, 5/2... = ...-2, -1, 0 , 1, 2...

In the next post, we will see how the discrete values of these propagating oscillations are manifest in the visible light spectrum, but I don't want you to choke by giving you too much at once. Also, I must mention that what we have developed in this post varies somewhat from Larson's development, which takes a slightly different approach to the photon model. However, as we have seen, the new approach has an explicit mathematical basis, whereas Larson's approach is only textually described. I hope the novel approach presented here using PAs, worldline charts and the mathematical system of numbers, adds value to his work.

As I mentioned at the end of the post of above, the development that I'm presenting differs somewhat from Larson's at the point of deducing the motion of the photon. In fact, in some respects at least, the two approaches are the inverse of one another. The best way to explain the difference is to use the 3D Clifford algebra Hestenes dubbed Geometric Algebra.

The history of GA is very interesting and I think everyone should read it. Historically, numbers have been the domain of algebra, and magnitudes the domain of geometry, which the ancient Greeks kept separated from numbers. The reason is that magnitudes in geometry have more information than numbers. There are magnitudes of lines, of planes, and of volumes. GA permits the union of magnitudes and numbers in a set of algebraic operations that is based on some deep mathematical relations known as Bott's periodicity that seems to form the basis of the Clifford algebras. GA happens to be the 3D version of these. I think there are 8 altogether, but I don't recall for sure.

Anyway, GA lends itself to the Reciprocal System, which assumes that the physical universe conforms to the three dimensions of Euclidean geometry. Given the marriage of numbers and magnitudes in GA, eight geometric magnitudes can be identified that can be associated with numbers in the basic operations such as addition, subtraction, and multiplication, and then there is the inverse operation as well.

There is an uncanny symmetry in the Clifford algebras. It can be seen in GA as the rank order of the 8 geometric magnitudes, which are classified by the number of their dimensions into groups dubbed blades. In the first group, the 0-blades, there is only one magnitude. It is called a scalar magnitude. In the second group, the 1-blades, there are three magnitudes. These are called vectors. There are also three 2-blades, called bivectors, and there is one 3-blade, called a trivector, or pseudoscalar. Notice the symmetry of the number of magnitudes in ascending blade order:

1, 3, 3, 1

This symmetry is present in all the Clifford algebras, but the number of groups (blades) is always n+1, where n is the number of dimensions of the algebra. Now, it just happens that the 8 groups in GA correspond to the types of motion in the RST as well. In the first instance, scalar motion has no specific direction, but is translational motion in all directions, corresponding to the GA 0-blade. In the second, there are three possible independent directions, or vectors, in which vibrational motion can take place, corresponding to the three GA 1-blades. In the third, there are three ways to combine the three independent vectors to form three rotations, corresponding to the three GA 2-blades, or bivectors. Finally,there is one way to combine all three vibrations, corresponding to the GA 3-blade, or trivector.

Apparently, Larson was unaware of Clifford algebras, or GA, but his development can be described in terms of GA's eight magnitudes. He starts with the 0-blade, the initial state that I describe with the PA in Part VII of my posts. It is the uniform space/time progression ratio of 1/1, one unit of space for one unit of time. Next, he introduces the "direction" reversals in this initial state, which leads to the second state, the 1/2 or 2/1 progression ratio, that I also show using the PA in Part VII.

However, for some unknown reason, at this point he skips to "another possibility," which he calls "simple harmonic motion." This can best be compared to the 1-blade, a one-dimensional oscillation, which he compares to the physical photon, because its composite motion consists of the 1-blade oscillation, or vibration and the two remaining dimensions of the initial scalar progression, so that, relative to a fixed spatial coordinate system, the vibrating entity appears to propagate in "one of the vacant dimensions;" that is, the "direction" reversals are effective in only one of the three dimensions of the initial scalar motion.

Using this initial vibration, a photon of unit frequency, Larson then "applies" a rotational motion to it, rotating it about its midpoint in the two orthogonal dimensions, which forms a two-dimensional rotation that can be visualized as "two, inter-penetrated, disks." This 2D rotational motion corresponds to two GA 2-blades, or bivectors, which can then be rotated one-dimensionally about the remaining third axis, forming the combination of magnitudes corresponding to the GA 3-blade, or trivector.

Thus, Larson derives a complete model of scalar motion that he subsequently uses to develop his Reciprocal System theories that are remarkable in their success at describing observed physical phenomena and resolving many outstanding problems in current theory. However, many have found difficulty in following and accepting the bonafides of the scalar motion model just described, in spite of the success Larson achieves with it.

The approach I have taken that I describe in this thread is an attempt to address these concerns. Its point of departure is precisely where Larson jumps to "another possibility" in his development. After deducing the second state, he jumps directly to the third state, from the continuous "direction" reversals of the second state, to the periodic "direction" reversals of the third state, without explaining how the third state emerges from the second state, as I have explained it, at least in general, in Part VIII above.

The effect this has had on Larson's model of scalar motion is dramatic, but it can easily be distinguished in the symmetry of GA. The difference is that Larson starts with the vectors, the 1-blades, and builds on this magnitude to the 2 and 3-blade magnitudes, whereas going the other way, I start with the 3-blade magnitude, and derive the 2 and 1-blade magnitudes from the 3-blade. However, I have no hope of ever carrying the consequences of this different procedure to the extent Larson carried the consequences of his procedure, which was truly remarkable and breathtaking in its scope. Nevertheless, I fervently hope to be able to clarify Larson's model of scalar motion and eliminate the troubles that are currently being experienced with it.

The basis of this optimism is founded in the symmetry of GA. I think going from the right to the left, instead of from the left to the right, will, in the end, not substantially change the results that he has achieved, while, at the same time, it could help us solve some of the current problems we see in the model.

The reason I point this out now is two-fold. First, I don't want to confuse anyone reading Larson's works for the first time. Second, I want to illustrate the main thesis of this thread: that Larson has introduced a new system of physical theory. He used his new system of physical theory to construct a general theory of the universe, the universe of motion. This is an outstanding achievement that will yet go down in the history of science, as a major milestone, if not THE major milestone, of modern theoretical physics.

More importantly, however, I want to show that the Reciprocal System is indeed a system that can be understood and used by others, even those as simple and naive as I am, to investigate the properties of the universe of motion and compare the results to the avalanche of observations now pouring in daily. What an exciting prospect.

In the previous posts in this series, we have seen how, with only the numbers of the natural number system, without the need of zero, magnitudes can be operationally interpreted in the rational number system, generating the integer number system, wherein zero becomes merely a reference symbol of the unit ratio, 1/1. In distinguishing the difference between the quantitative and the operational interpretation of numbers, we've used the word magnitude interchangeably with the word number. However, the ancient Greeks were careful to keep the concept of number separate from the concept of magnitude. Evidently, to them, numbers were for counting objects, while magnitudes were for measuring objects, two different tasks; that is, they used the system of numbers for counting quantities and another, separate, system of ratios of numbers for measuring magnitudes. As one author puts it: [6]

Of course, the difference between the two systems is found in the additional information of "direction" that the concept of magnitude incorporates; that is, while 1/2 and 2/1 are equal to the same number, 1, they are distinguished by their inverse "directions," and thus constitute two distinct magnitudes. Certainly, this information is most important in determining the balances of the relative weights of trade goods and the current state of ongoing financial transactions. Thus, we are led to a dual system of weights and measures, but we see that the concept of weights is related to the concept of measures through the concept of "direction." However, the concept of "direction" can have three fundamental orientations in the three-dimensional world of geometric measures, an thus we come to the concept of "direction" in three orientations, which leads to the principles of the 3D geometry of space measures, and, ultimately, to the principles of the 3D calculus of space and time, or motion, measures.

Now, however, Larson posits that objects, i.e. matter, are nothing but motion, i.e. space/time ratios, and we have seen how, given a universal progression of space and time, if reversals in the "direction" of the space or time progression occur at a given point, the progression of space, or time, at that point ceases to increase uniformly, but instead "oscillates" between two space, or time, points in the progression continuously, effectively halting the progression of space, or time, at that point and, in the process, creating a discrete, or independent, unit of motion. While this textual description of a physical abstraction is fairly complex, it is easily understood in terms of the PAs and the worldline chart of figures 1-5 above. Clearly, though, the concepts of numbers and magnitudes, and their relationship through the concept of "direction," which is so plainly manifest in the development so far, need to also incorporate the concept of orientation, or the three dimensions of Euclidean geometry, to be complete.

It is important to note, however, that the fact that the course of the development to this point has not included the concept of dimensions is a demonstration that this theory of motion is background-free. In other words, it has been shown that the consequences of scalar motion, from the initial state of uniform progression, the unit space/time ratio, to the second state of non-uniform progression, leading to the two non-unit space/time ratios, and the subsequent combination of these two ratios into a composite space/time ratio, constitutes a description of discrete space/time ratios with purely scalar relationships; that is, they have been completely defined as non-oriented magnitudes of scalar motion. Nevertheless, the Reciprocal System assumes that the discrete units of motion, of which the universe of motion is composed, exist in three dimensions. So, now we have to see how this aspect of the universe of motion is manifest.

(continued in following post)

Since, in the universe of motion, nothing exists but space/time, or motion, this means that space and time can have no independent existence as space or time apart from motion. Nevertheless, we are accustomed to thinking that the distance between two objects constitutes "space" and that this "space" is defined as a set of locations that satisfies the postulates of Euclidean geometry, or in modern times, even the postulates of non-Euclidean geometry. Indeed, it has been shown that we can also think of the time interval separating two events as "distance," and incorporate this notion into the idea of "space," forming the concept of four-dimensional "spacetime" that is employed in general relativity theory to explain the motion of gravity, as something due to the warping of the locations of "spacetime," and thus effecting the motion of matter "in spacetime."

Clearly, though, the distance between two objects cannot exist, unless a motion has, or is, or will, ocurr between them. Hence, the "space," or distance, separating two objects was, or is, or will be, created by the relative motion of the objects. Furthermore, if the motion separating a set of objects is all in one direction relative to the objects, the separation, or "space" between them will be in one direction. If the motion is in two directions, the "space" will have two directions, and if the motion is in three directions, the "space" will have three directions. In fact, viewing "space" this way, we can reinterpret Euclid's fifth axiom, that there is only one line on a point not on a line parallel to it, as the definition of one-dimensional motion; that is, any motion that is not in one dimension, necessarily must be motion in another dimension.

Since the universe of motion is limited to three such dimensions, we see that it is possible that the descrete units of scalar motion, once they exist, can be separated in three directions. This means that, if we consider an S unit in figure 5 above, as occurring at point B in the scalar space/time progression, subsequent to the occurrance of a previous unit at point A in the progression, and before a similar unit at point C in the progression, the question of their relative spatial positions depends on the direction of the space/time progression that separates them. However, the space/time progression is scalar, which means it has no specific direction. Therefore, if we decide that units A, B, and C all lie in a straight line, we have implicitly assigned one dimension to the space/time progression, but doing so contradicts the assumption, contained in the first postulate of the Reciprocal System, that the motion exists in three dimensions. If the assumption were that discrete units of scalar motion exist in one dimension, then units A, B, and C, would necessarily lie on a line; if the assumption were that they exist in two dimensions, then they would have to lie in a plane, though not necessarily on a line; but with the assumption that they exist in three dimensions, they can lie on a line, in a plane, or neither of these, but anywhere in a volume.

However, discrete units of scalar motion occupy the interval of space, or time, defined by the "direction" reversal of the space or time progression occurring at that point, and, since a scalar increase from that point is three dimensional by definition, there seems to be an apparent contradiction between the concept of one "point" in the progression, and a subsequent "point," because a 3D progression necessarily expands as a volume from point to point in the progression. Thus, point A in the progression would be smaller than the subsequent point B, and B smaller than subsequent point C, ad infinitum. Nevertheless, because the units, A, B, and C, consist of only one interval of the progression; that is, they consists of a decrease/increase from the point A, B, or C, to the point A-1, B-1, and C-1, and back again, the direction of the scalar increase of space/time that occurs between them is not defined before the "direction" reversals in the progression that defines them occurs. In other words, their relative positions are not defined by the progression until they come into existence at the point in the space/time progression that the "direction" reversals first commence. Since any position on the sphere of expansion defined between the units, by the space/time progression interval, equals the scalar magnitude of the interval between these units, one position is as likely as any other and the actual position is random on the surface of the sphere. In fact, since the relative direction thus determined is redetermined at each unit (i.e. from a sphere centered on the previous location), the relative locations of units A, B, and C, are completely random.

This emergent feature of the theory is so similar to the observed behavior of quantum phenomena that it immediately provokes the intriguing thought that it could relate to the long sought foundation of that mysterious aspect of reality. Only time will tell, however, but we now can see how that a spatial coordinate system (and the inverse, a temporal coordinate system) can be used to define the "space" separating discrete units of motion in the universe of motion, yet no "space" as a background structure actually exists in which these units are "contained."

If you haven't already done so, it might be a good idea to pause and reflect on the significance of what has just been achieved here. Billions of dollars are currently being spent on trying to find a way to do it within the context of the Newtonian system of physical theory, but it appears that it can only be done under the new Reciprocal System of Physical Theory. This claim should at least get it a fair hearing in the halls of mainstream physics, but, alas, I doubt that I will live long enough to see it. Indeed, maybe, none of us will.

References:

6) John L. Bell, The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development, Kluwer, 1999.

Now, that we have theoretically established discrete units of motion, existing in three dimensions, with two reciprocal aspects, space and time, from nothing but motion, as necessary consequences of the fundamental Postulates of the Reciprocal System, we need to look at what we have. Referring to the worldline chart in figure 9 above, we see depicted three theoretical entities, the S unit, the T unit, and the ST unit, which is also a TS unit, depending upon the point of view that one assumes. However, the important thing to keep in mind is that all these units are units of motion, and that the worldline chart itself is a plot of the space/time progression. Recall that, as space progresses in the chart, or increases, horizontally, time increases vertically. Therefore, a plot of the unit space/time progression ratio, where space and time progress equally, creates a 45-degree line dividing the progression into two sectors. Plotting a location where space does not appear to increase, but time continues to increase normally, creates a vertical line, while plotting a location where time does not appear to increase, but space continues to increase normally, creates a horizontal line on the chart. Of course, neither space nor time can stop progressing, but the same effect is achieved by the continuous "direction" reversals in the progression of one aspect or the other of a discrete unit of motion. It's as if one aspect or the other of the scalar motion were "marching in place," so to speak.

The "direction" reversals create a space/time progression ratio of s/t = 1/2, when the space progression is continuously reversing, and a space/time progression ratio of s/t = 2/1, when the time progression is continuously reversing, as indicated by the chart positions of the S and T units respectively. Also, as we have seen, combining S and T units produces a combination unit designated ST, where, because both the space and the time progression are reversing "direction" in the constituent S and T units of the combination unit, while together they form a unit progression ratio at the same time, a composite ratio results that progresses at the speed of light relative to uncombined S and T units, which are stationary in space or time. We identified this combination unit with the physically observed photon. A summary of all this information is contained in table 1.

http://www.rstheory.com/member_files/NZStates.jpg

Table 1. RST Initial States of Scalar Motion

As previously mentioned, Larson’s development differs somewhat from the above. His approach is based on the same principles of scalar motion, wherein the “direction” reversals lead to space and time displacements, as we have done, but his development does not combine the S and T units to form the ST (TS) units, as we have done in row 4 of the table above. Primarily, this is due to a different interpretation of the geometric dimensions of the progression ratios in rows 2 an 3 of table 1. Whereas, the new interpretation is that the geometric dimensions of these progression ratios, 1/2 and 2/1, result in 3D S and T units, since they are scalar motion ratios, and scalar motion is, by definition, motion in all directions, Larson, on the other hand, interprets these as 1D units of scalar motion.

On this basis, the left PA in row 4 of table 1, representing a space/time progression ratio of s/t = 2/3 = -1, is interpreted as an oscillation in only one dimension, leaving two dimensions of motion undisplaced at the unit progression ratio. As the total time displacement is increased in this unit, which we will designate S1, an oscillation of lower and lower frequency is produced in one-dimension, providing theoretical units of varying frequency, but all of which propagate at light speed, relative to the S units, in an orthogonal dimension perpendicular to the unit's vibration. However, the propagation at light speed is due to the fact that one or the other of the two remaining scalar dimensions in the S1 unit are not displaced. This is the theoretical entity that Larson identifies with the physically observed photon. A similar treatment of the s/t = 2/1 progression ratio leads to a T1 unit.

However, the S and T units, as stationary in space and time respectively, don’t exist at this point in Larson’s development, since, to remain stationary in space, or time, they would have to have three dimensions of displacement, as a space/time unit progression ratio in any vacant dimension, would result in propagation, or progression, of the unit in one of the vacant dimensions. Thus, Larson begins with a T1 unit, a high-speed, or high-frequency, photon, illustrated by the right PA in row 4 of table 1, representing a space/time progression ratio of s/t = 3/2 = 1, and adds a time displaced unit to it, i.e. a 2D space motion, in the form of a rotation, which rotates it about its midpoint, in two orthogonal dimensions, forming a unit that can be envisioned as two orthogonal, interpenetrated disks, as shown in figure 10.



Figure 10. Larson's 2D Rotation of 1D Vibration

Consequently, a rotating 2D S unit, S2, combined with a 1D oscillating unit, T1, forms a rotating unit that is what Larson calls “the rotational base,” an S2T1 unit that has no net space or time displacement; that is, this combination of scalar motion is s/t = 1/2 + 3/2 = 4/4 = 1/1 = "0". Another way to look at it is that a two-dimensional -1 unit of scalar motion, plus a one-dimensional +1 unit, equals a three-dimensional unit with zero net displacement; that is, -1 + 1 = 0. The 2D unit is equal to -1 because, at unit value, 1 squared, or 1^2, is equal to 1, and the minus sign simply indicates the negative "direction" assigned to the 1/2 magnitude of the space progression ratio.

Larson then uses the rotational base as the foundation of subsequent combinations that have increasing units of time and space displacement in one and two-dimensional increments. Readers interested in the details of how he does this can consult his works for additional information. [7] However, for our purposes, it is sufficient to note that the major difference in the two approaches should be regarded as the difference between a bottom-up approach, wherein Larson combines units of one-dimensional and two-dimensional scalar motion to obtain combinations of three-dimensional units, and a top-down approach, wherein, in contrast, we start with the initial three-dimensional units, S and T, and combine them to obtain combinations of one and two-dimensional units, as seen, for instance, in the ST (TS) combo in table 1.

In the final analysis, therefore, the result is the same: discrete units of scalar motion, combinations of discrete units of scalar motion, and relations between discrete units of scalar motion with one, two and three space/time dimensions, are used to construct a theory of a universe of motion. Nevertheless, the full implications of the new approach have barely begun to be explored. For this reason, in these posts, Larson's development will be regarded as the baseline theory, with the understanding that the new development possibly may address some issues that arise in the course of the discussion more successfully than Larson's development is able to do.

So, in subsequent posts, we will begin to examine the development of theoretical photon units and see how their properties emerge, and then we will proceed to develop theoretical subatomic units such as electrons, positrons, neutrinos, protons and neutrons, and eventually full atomic units, and see how the basic properties of matter emerge from these, such as gravitational and inertial mass, leading to mechanical phenomena and Newton's laws; and properties such as heat and temperature, leading to the phenomena of the gas, vapor, liquid, and solid states of matter and Boyle's law, Boltzman's constant and others; magnetic and electrical properties, leading to electrical and magnetic phenomena, such as electrical charge, electrical current, ferromagnetism, magnetic induction, electromagnetic waves, and Maxwell's equations; and finally, the properties of atomic weight and isotopic mass, leading to phenomena of atomic spectra and radioactivity, and a new, non-nuclear, model of the atom, leading to the bizarre behavior of quantum mechanics.

Of course, we will only be able to cover the essentials of these topics, many of which are not even fully developed yet, as the new system has barely begun to be employed at this point, but, hopefully, we can show enough to provide a taste of the Reciprocal System's capability to do, from first principles, what the LST cannot do, and, further, to demonstrate that, whereas current LST science must rely on a diverse collection of different theories, specifically tailored to meet the needs of explaining the above list of phenomena, the RST theory of a universe of motion is a unified whole, based on nothing but motion.

References:

7) Dewey Larson, The Structure of the Physical Universe, in Three Volumes, http:\\www.rstheory.com

I'm working on Part X of this series and expect to have it completed sometime this weekend, but by sometime next week, I'll be out of the country for a week to 10 days, and probably will not be able to participate in the forum during that time (unless cheap European hotels have Internet connected computers available to their guests).

So, if anyone has a comment or question that they would like answered before I leave, it would be a good idea to post it soon. The new Part 10 will show that the dimensions of LST physics such as mass, momentum, acceleration, force, energy, etc. can all be expressed in space/time terms and that these formulations actually conform to the mathematical entities of the fourth Clifford algebra in an unexpected, and startling fashion.

Regards,

Excal

Wow, I like this thread. I'll keep my mouth shut since I'm obviously out of my league, but I'm fascinated by this idea. I'll certainly be reading the thread.

As I have explained in this series of posts, the fundamental difference between Larson’s RST, and the LST, is the definition of motion. In the LST, motion is defined as a change of an object’s position, or the evolution of a field, or wave function, as a function of time, x(t), defined in an inertial reference system of space and time, whereas, in the RST, motion is defined simply as the universal change (progression) of space and time that is prior to all things, including physical entities with properties of radiation, matter and energy. Hence, the RST adds a new type of motion, called scalar motion, to the LST concept of motion, called vectorial motion, which enables the LST concepts of mass, force and acceleration, etc, to be formulated in terms of the more fundamental units of space/time. Moreover, the new concept of scalar motion enables a clear distinction to be made between the two types of motion that eliminates the need to regard space and time as a background structure, which all LST definitions of motion require.

For instance, we see in the background-free RST, that matter and energy magnitudes are related by velocity magnitudes because they are all scalar motion magnitudes of different dimensions (degree), as explained below (the delta symbol, ∆, is omitted from the following expressions, but the s and t variables should be understood as ∆s and ∆t.)

In the RST, mass has the dimensions of three-dimensional motion,

(s/t)^3,

while the resistance to this motion, inertial mass, has the inverse of these dimensions,

(t/s)^3.

Momentum is mass times velocity and has dimensions

p = mv = (t/s)^3 * (s/t) = (t/s)^2.

Acceleration is velocity per unit of time with dimensions

a = (s/t)/t = s/t^2,

and force is mass times acceleration with dimensions

f = ma = (t/s)^3 * (s/t^2) = (t/s^2).

Energy is the inverse of one-dimensional velocity with dimensions t/s, so the energy of mass is the mass times velocity squared, or

E = mc^2 = (t/s)^3 * (s/t)^2 = t/s,

while the energy of radiation is Planck's constant times frequency, or

E = hv = (t/s)^2 * s/t = t/s.

However, notice that, here, the dimensions of h are t^2/s^2, instead of t^2/s (energy multiplied by time), and the dimensions of frequency are s/t, instead of 1/t. This change in the dimensions relating energy to radiation is a result of the fact that all motion, even the oscillating motion of frequency, has space/time dimensions. In the case of the photon, the oscillating motion involves one unit of space traversed repeatedly. Since the perplexing dimensions of Planck's constant, units of energy multiplied by time, or units of momentum multiplied by distance, are the basis for Heisenberg's uncertainty principle, this simple resolution of the enigma has important implications for quantum mechanics.

This is only a small sample of the physical quantities that can now be expressed in dimensions of space and time, but it's enough to show that this is a very significant addition to all that has been accomplished in the LST. If it is valid, it carries wide-ranging implications for theoretical physics. Given its simplicity, one wonders why it has never been suggested before. Larson attributes this oversight to a preoccupation with multiple dimensions of space that has excluded a consideration of multiple dimensions of motion. He writes: [8]

However, this is only true when we regard motion as the one-dimensional time rate of change of an object’s position, or the time evolution of fields, or wave functions. When we regard motion as the reciprocal relation of space and time, it is not true that multi-dimensional motion cannot be represented in the reference system. In fact, something marvelous is now available that is the mathematical equivalent of multi-dimensional motion. Recall that Hestenes credited William Clifford as perhaps “the first person to find significance in the fact that two different interpretations of number can be distinguished,” a significance that plays a crucial role in the mathematics and physics of the RST, in that it enables us to see that the natural number system is related to the rational number system by the ratio operation, and that the integer number system is generated by the rational numbers system, integrating these three number systems into one whole that removes the logical difficulties of negative numbers, and even the mystery of the imaginary number “i,” although I may not have explained the latter point in these posts.

It turns out that Clifford did much more than provide us a way to understand the “negative” and “positive” integers, with the quantitative/operational distinction of the interpretation of number, which enables us to include the concept of “direction” with the concept of quantity, in the properties of numbers. Actually, he went beyond this and added yet another property to the concept of number, thus generalizing it even more. This additional property is the property of dimension. While the property of dimension is a routine property of mathematical expressions in the sense that an n-dimensional quantity is one that takes n independent numbers to define it, its use can be confusing. For instance, we say that a line is one-dimensional, an area is two-dimensional, and a cube is three-dimensional, because they extend in one, two, and three spatial directions respectively.

However, if we say that a number is n dimensional, we don’t mean that it extends in n dimensions of space. What we mean is that the number defines n magnitudes. Thus, the number n^x is x number of magnitudes, or a magnitude multiplied by itself x number of times, but if the magnitudes of an n-dimensional quantity are unequal, it requires separate numbers to define each magnitude. For instance, the “space” of the conventional coordinate reference system is three-dimensional; that is, it takes three separate numbers to define a location in the reference system, x, y, and z, but a location itself is not three-dimensional. The numbers simply specify the location’s position relative to the origin by specifying the magnitude of its distance from the origin in each of three orthogonal directions. Hence, motion in this system can only be one-dimensional, because it takes two sets of three numbers (triplets) to specify a change in location, i.e. (x, y, z) – (x’, y’, z’). Therefore, in order to represent multi-dimensional motion, we need to have additional dimensions of “space,” but we don’t observe any evidence of these extra dimensions of space in the physical universe.

Nevertheless, there is another approach available, when we abandon the notion that motion is necessarily defined as the changing position of an object over time. We can see from the equation of motion, v = ∆s/∆t, that an object’s position is not required in its definition. The only requirement is that a delta, or a certain change, in the quantity of space exists for a given change in the quantity of time, a time rate of change of a quantity of space. Clearly, then, if we assume that space and time are quantized, the equation of motion is satisfied by an increase or decrease in the units of space for each increase in the units of time of a given physical situation, regardless of the particulars of the physical situation, as long as the change in space and time are defined.

For instance, the expansion of a line of railroad cars from one to twenty over a period of twenty minutes could be described as a 1D motion. Likewise, the expansion of a meatball into a meat patty could be described as a 2D motion, and the expansion of balloon could be described as a 3D motion. In each of these cases, the equation of motion is satisfied by a scalar increase in the number of material units over a number of time units, in a given dimension. In the case of the 1D motion of railroad cars, the increase is in the number of railroad cars, or units, in a line. In the 2D case, it is an increase in the number of meat units in an area, and, in the case of the balloon, it is an increase in the number of gas units in a volume. The point is, though, in each case, the increasing units of space that these objects occupy define an n-dimensional motion as time increases. And, of course, this n-dimensional motion can be represented in the conventional reference system as an n-dimensional expansion, and if we assume that the physical universe is composed of a universal motion, we then assume that a universal increase of space is just as much a part of reality as is the familiar universal increase of time, since we have defined motion as the reciprocal relation of space and time, or the magnitude of a rational number, operationally defined. Therefore, we conclude, that an n-dimensional, rational, number, consisting of two natural, or scalar, numbers, operationally interpreted, is equivalent to an n-dimensional scalar motion in the conventional reference system, representing an n-dimensional increase in the number of space units for a given increase in the number of time units.

The question is, though, how do we define an n-dimensional, rational, number? As we have seen, the operational interpretation of a rational number expresses the integer “displacement” from the unit ratio in two “directions,” which adds the concept of “direction” to the natural number concept of quantity, transforming it into a magnitude, but this magnitude is a ratio of natural numbers. Introducing dimensions into this definition of number, as a magnitude, is tricky, because it might affect the integrity of primary operations of arithmetic such as addition, subtraction, multiplication, inversion, etc, and the algebraic distributive and associative properties of the operations. Fortunately, however, the genius of young William Clifford was up to the task. His Clifford algebras meet this challenge of further generalizing the concept of number by adding the concept of dimension to the concept of quantity and “direction” that we already have, while guaranteeing that the integrity of the primary mathematical operations and their algebraic properties are preserved.

(continued in the following post)

To understand how this is done, it’s helpful to view the multi-dimensional expansion of the number concept as the dimensional expansion of the two “directions” of magnitude as shown in figure 10:



Figure 10. Clifford Algebra Binomial Triangle

where n is the number of the algebra designated Cn. To demonstrate the significance of this in relation to the 3D spatial reference system, or Euclidean geometry, a definition of the first four Clifford algebras, C0-3, in terms of the quantitative/operational interpretation of the number concept, follows. (Note: For the sake of brevity, I will refer to this generalized number concept as the integrated number system, or INS, from this point on.)

1) C0 is the one-dimensional algebra of natural numbers; that is, the numbers of this algebra have one property that can be used to define a value of quantity (scalar value). Geometrically, this corresponds to the concept of points.

2) C1 is the two-dimensional algebra of rational numbers, existing in one dimension; that is, the numbers of this algebra have two properties that can be used to define a value of quantity plus a value of “direction” (vector value), which leads to the set of real numbers including complex numbers (a line with two “directions”). Geometrically, this corresponds to points and lines.

3) C2 is the three-dimensional algebra of rational numbers, existing in two dimensions; that is, the numbers of this algebra have three properties that can be used to define a value of quantity, and two values of “directions” (bivector value), which extends the set of real numbers to include quaternions (an area with two “directions”). Geometrically, this corresponds to points, lines and planes.

4) C3 is the four-dimensional algebra of rational numbers, existing in three dimensions; that is, the numbers of this algebra have four properties that can be used to define a value of quantity, and three values of “direction” (trivector value), which extends the set of real numbers to octonions (a volume with two “directions”). Geometrically, this corresponds to points, lines, planes, and volumes.

Notice how that the “direction” property is conserved here, in that in C1 the two “directions” are “directions” of a 1D line, but in C2 the two “directions” are “directions” of a 2D area (bivector), and in C3 the two “directions” are “directions” of a 3D volume (trivector). Mathematically, this is very different from two “directions” along a line (vector), or two “directions” along two lines (two vectors), or two “directions” along three lines (three vectors), which would ordinarily be thought of as three abstract orthogonal axes in terms of which points in lines, planes, and volumes can be defined, and the magnitude and direction of the motion of objects can be defined.

Instead, the conservation of the “direction” property in these algebras requires the existence of mathematical entities, or numbers, of zero, one, two and three dimensions; that is, the two C1 “directions” are a 1D expression of two “directions;” the two C2 “directions” are a 2D expression of two “directions;” and the two C3 “directions” are a 3D expression of two “directions.” In other words, in the dimensional transition from C0 to C3, the properties of the numbers in the INS are transformed dimensionally. That is to say, the numbers themselves are transformed from a scalar number with no “directions” in C0, to a line number in C1, to a plane number in C2, and to a volume number in C3, with each dimensional number having two “directions.” Moreover, each of these algebras contains all the properties of the previous algebra, so that C3 contains the n-dimensional numbers of C0-3.

Thus, C3 defines a set of 0, 1, 2 and 3 dimensional numbers for which the primary arithmetic operations such as addition, subtraction, multiplication, and division, and the algebraic properties such as associative and distributive properties are preserved. This is a type of algebra known as associative algebra.[8] All of the Clifford algebras are associative algebras, but the C3 algebra corresponds to the three-dimensional Euclidean space and for this reason is known as the G3 algebra, called Geometric Algebra (GA) by Hestenes. [9]

In GA, the two “directions” of the INS are known as “orientations,” and they are interpreted as “directed line segments,” “directed areas,” and “directed volumes.” However, true to the geometric roots of mathematics, these are concepts of space only, and in the LST applications of GA to physics, these multi-dimensional space concepts are treated as space/time structure; that is, they are used to describe 1D motion in a line, or about a line through an area, or about an area through a volume, etc. in one, two and three dimensions, and in a coordinate-free manner. However, the concept that these directed magnitudes might be treated as one, two, and three-dimensional magnitudes of motion is not considered, precisely for the reason noted by Larson.

Nevertheless, as we have seen, the correspondence of the RST definition of scalar motion with the INS actually identifies these multi-dimensional numbers with the multi-dimensional motion of the Reciprocal System. One way to appreciate the significance of this is to consider how the sciences of mathematics, geometry, and physics, are related. Fundamentally, the science of mathematics is the science of numbers, while the sciences of geometry and physics are sciences of magnitudes. The generalization of the concept of number to incorporate the concepts of magnitude, i.e. direction and dimension, has been a difficult, centuries-long, struggle to apply mathematics to geometry and physics.

However, the magnitudes investigated in geometry are the magnitudes of space, while the magnitudes investigated in LST physics are the magnitudes of space and time as motion, in space and time as a container, and as we have discussed in these posts, the magnitudes of this motion are defined as a 1D motion in which the position of an object, or the properties of a field, or the probabilities of a wave function, change, or evolve, over time.

This means, then, that there is a fundamental disconnect between the meaning of the concepts of direction and dimension in mathematics, geometry and physics. In geometry, the meaning of direction is synonymous with the meaning of “direction” and dimension of the numbers in the INS; that is, a point has no directions, a line has two directions (opposite ends), a plane has two directions (opposite sides), and a volume has two directions (opposite surfaces). Each of these can then be oriented in three, orthogonal, dimensions.

In contrast, the meanings of direction and dimension, in LST physics, are not synonymous with the meaning of “direction” and dimension of the numbers in the INS. Instead, the meaning of direction in physics means direction of space/time (motion) in “space,” as defined by points that take three independent variables of space magnitude to define them, which is the only meaning ascribed to multiple dimensions in the LST. Thus, in LST physics there are an infinite number of directions that may be defined as space/time magnitudes (motion) with two “directions,” either away from, or towards, a given point defined by three dimensions. Hence, the types of space/time (motion) magnitudes in LST physics are defined in terms of the meaning of direction and dimension as they apply to a point, not as they apply to the magnitude of the motion, as an entity; that is, LST motion is defined relative to the three dimensions of a point, rather than the three dimensions of a space/time magnitude, and the two possible “directions” of this motion are defined relative to the point, not the “directions” of the space/time magnitude. Consequently, four types of motion are defined in the LST:

1) Translation, which is motion in an outward “direction” or an inward “direction” toward a single 3D point along a given path (direction).
2) Vibration, which is both an inward and outward motion relative to two points.
3) Rotation, which is neither outward nor inward motion toward a point, but is motion around a point.
4) Vibrational rotation, which is a combination of a vibration’s inward/outward motion relative to a point in combination with rotation’s motion around the point.

Thus, translational motion defines a point (a target); vibrational motion defines a line between two points; rotation defines an area about a point; and vibrational rotation defines a volume within an area of rotation (a cylinder). However, the point is that these are magnitudes of space, not magnitudes of space/time. In the LST, space/time magnitudes are always one-dimensional, while in the INS, geometry, and RST physics, the respective magnitudes of number, space, and space/time are all multi-dimensional and show a one-to-one correspondence that is missing in LST physics. This difference can be illustrated and compared in a chart like that of table 1 below.



Table 1. Magnitude Direction and Dimension Analogs

Here we see a one to one correspondence between the magnitudes of the multi-dimensional numbers of the INS and the space magnitudes of geometry, while there is no such correspondence between the INS magnitudes and the LST magnitudes of space/time (motion). Clearly, this is due to the fact that the magnitudes of space/time are confined to one dimension in LST physics. In contrast, we see the same correspondence between the INS numbers and the RST space/time magnitudes, as we see between the INS numbers and the geometric space magnitudes.

In the final analysis, what this means is that it is now possible to develop a new physics based on multidimensional space/time magnitudes.

References:

8) Wikipedia Article, "Associative Algebra," http://en.wikipedia.org/wiki/Associative_algebra

9) Wikipedia Article, "Geometric Algebra", http://en.wikipedia.org/wiki/Geometric_algebra

Over at Crooked Timber, there is a lengthy discussion going on over a New York Times' article by Brian Green entitled "About Einstein’s famous equation E=mc²". Green reveals in the article that the modern LST view that mass and energy are equivalent means that mass is consumed in chemical reactions that release energy, which is disconcerting to many people, since it’s difficult to think of mass varying with energy like the potential energy of gravity, or a wound up spring, or the heat from the sun. John Quiggin, the author of the blog, writes:

So far, 76 replies have been posted on CT, and the discussion continues to echo on other blogs such as cosmicvariance. Some of these support Green’s conclusion, and others refuse to accept it. Most comments point to the tiny amount of mass-energy consumed in chemical reactions, but a few discuss the real issue:
In reply, a practicing physicist writes:
The hostility of text editors to mathematical notation makes it difficult to use equations on blogs, and in forums, but the point made in these comments above are clear, even if the notation is not: dimensionally, there is something wrong with equating mass and energy. Larson makes the point that if mass can be converted to energy, and energy to mass, then the unavoidable conclusion is that neither of these two quantities can be fundamental, because that which can be changed into something else, cannot be fundamental by definition. There must be something more fundamental that they both have in common.

Of course, in the universe of motion, the fundamental entity common to radiation, matter, and energy, is motion. However, as we have seen, while v = s/t is a magnitude of motion, so is v = t/s, and at the speed of light they are the same value, v = s/t = t/s = 1/1 =1. The difference arises when one of the quantities in the equation of motion exceeds unity. In one “direction,” 1/n, lie all the velocity based physical phenomena. In the other “direction,” n/1, lie all the energy based physical phenomena. Without this understanding of the INS mathematics, confusion ensues.

On this basis, however, Einstein’s equation, relating matter to energy, is easily understood: mass is not equivalent to energy. It is the three-dimensional motion that resists the three-dimensional motion of matter. Energy is the inverse of one-dimensional velocity. Therefore, to relate one to the other, this dimensional difference, the square of velocity, must be accounted for:

E = mc^2 = t^3/s^3 * s^2/t^2 = t/s.

We see the same thing in relating radiation and energy too. Radiation is not mass, it is velocity, but the velocity is a rotation, a frequency, which means that it can be expressed as revolutions, or cycles, per unit of time, 1/t. However, doing so obscures the fact that it is still a velocity, and that it cycles over one unit of space. Therefore, it ought to be expressed in its space/time dimensions, s/t, not 1/t. Hence, the dimensional difference between energy, t/s, and velocity, is energy squared, and must be accounted for in any conversion of one to the other:

E = hv = t^2/s^2 * s/t = t/s.

What we need, then, are the natural units of velocity and energy, s/t and t/s. As we have seen in Part V above, Larson used the Rydberg constant and the speed of light to arrive at the natural units of space and time, 4.558816 x 10^-6 centimeters and 1.520655 x 10^-16 seconds, respectively. The unit of acceleration, s/^t2 is then easily calculated as 1.971473 x 10^26 cm/sec^2, and, of course, the unit of speed is 2.997930 x 10^10 cm/sec. These velocity-based units are the units used in the LST, and thus are recognized, but the dimensions of the natural units of energy are the inverse of the velocity dimensions and are not recognized in the LST. Instead, the LST bases these relations on an arbitrary unit of mass, the gram, and to determine the ratio of natural units to this arbitrary unit, a physical measure of mass in conventional units is needed. Therefore, Larson chose Avogadro’s constant for this purpose. He writes:
Based on these three units, Larson summarizes both the velocity-based units and the energy-based units as follows:

http://www.rstheory.com/member_files/NaturalUnits.jpg

Table 2. Larson's Calculations of Natural Units of Primary Quantities

References:

10) Dewey B. Larson, "Nothing But Motion," Chapter 13, ISUS Web Site

Well, I'm back. Went to Venice, Milan, Geneva, and Paris. While I was trying to be a tourist, my heart wasn't really in it. I really missed having access to the Internet and being able to discuss physics. I expected to find some questions and/or comments when I got back, but it's very quiet in this thread. I don't know what to make of that yet.

The Einstein special aired on PBS while I was gone. I hope to be able to catch it when it airs again, but I went to the website and listened to some top physicists/cosmologists like Sheldon Glashow, Alan Guth, Frank Wilczek, and Michio Kaku, to mention just a few, comment on the E = mc^2 equation. They all have different takes on it of course, but most have to do with idea that the relation of mass and energy is unexpected, astonishing, or intriguing. However, I liked Nima Arkani-Hamed's comment the best. He is struck with the huge difference a simple and seemingly innocuous assumption can make in our world view:

I agree with Nima's assessment of the real lesson here: we need to recognize that the profound impact of our most basic and simple assumptions, whether they are correct or incorrect, makes an almost unfathonable difference in our world view. As I have been pointing out, the most basic of these assumptions in physics are our assumptions about the nature of space and time and how they relate to one another. The consequences of realizing that they are only the reciprocal aspects of a universal motion changes everything we think we know about the universe.

When I was traipsing about Europe, visiting the grand monuments of ancient civilization and the works erected by the economic powers of the time, watching the hoards of tourists from all around the world jostling and shuffling from one to the other on jetliners, trains, and buses, to snap photos, and to point out to one another the endless figures of gargoyles, gods, angels, men, women, children, and animals sculpted into the stone facades and gigantic columns of palaces, cathedrals, and public buildings and venues, I was struck with the awesome political and social implications that one can infer from these things. For good or ill, ideas are the most powerful things in existence. I felt a tremendous sense of gratitude that I have been blessed to view a universe that is so clear and logical, and one mostly free of the dark and oppressive assumptions of the past.

I reflected again and again on how the equations of Maxwell led to the harnessing of the electrical and magnetic propeties of matter, leading to a new civilization, with a new and fundamentally different economic foundation, powerful enough to enable millions of people from all nations to congregate from every direction, and to fly across the the oceans and the continents in a few hours, in comfort and ease, while tracking their progress on GPS connected computers, instantly communicating around the world in a myriad of ways, and not even having to loosen the latchets of their shoes in the process.

Then I realized how primitive it all is, really. The jet engines, as powerful as they are, shoot hot gas out the back, and the craft's mighty wings are spread to catch the power of the moving air and lift it into the air. If these are so great, what will the consequences be of equations that enable us to harness the gravitational property of matter? Clearly, the changes will usher in a new civilization that is as advanced relative to ours, as ours is to that of distant centuries. The thought that the beginning of those equations are even now being discovered, just blew me away.

Greetings Everyone,

Sorry for disappearing for several weeks. I had to tend to some pressing business in association with the ISUS conference which has just concluded. The good news is, however, that I prepared a powerpoint presentation for the conference, which I can now share with you. I think it will help to have pictures and graphs to make these reciprocal system concepts of number and their relationship to the Reciprocal System of Physical Theory more clear.

The presentation is entitled "The Reciprocal System of Mathematics" and is located here.

Earlier in these posts, I pointed out that these ideas were part of an ongoing research effort called the NZP, which means that the conclusions don't necessarily represent the views of ISUS and are at variance with some of those in Larson's works. The same disclaimer is repeated in the slides and emphazised once again here.

With the basics in place, we should probably move on to the cosmological implications of the RST, since that's what most of us in this forum are interested in. As we have seen, Larson developed a Reciprocal System theory, from the Reciprocal System OF Physical Theory. Since we refer to the fomer as the RST, we will refer to the latter as RSt, to differentiate the theory from the system it was developed under. As others are now striving to work on the RSt, and it's anticipated that many more will follow in the future, it's important to realize that, while there might be various RSts, there can only be one RST; that is, for a candidate theory to qualify as an RSt, it must not be justified on the grounds of a modified RST, which would be like changing the rules of the game, in the middle of it. If one wants to play a different game, then the rules can be modified, but then the consequent theory will be recognized as a coming from a different game.

Why is this so important? The reason lies in the power of the fundamental postulates to produce a theoretical universe of motion. If an aspect of that universe cannot be accounted for, or an aspect of the observable universe conflicts with the theoretical universe in a way that cannot be resolved, the fundamental postulates, upon which the universe of motion are based, would be falsified, which is the position we want them to be in. Changing them to accommodate a new conclusion, or new theory, is like changing the rules of the game, in the middle of the game. It invalidates the previous play and requires that the game begin anew.

We are vey happy with the results of the game to this point, and we don't want to start over (it's been 45 years since Larson published the preliminary edition of his theory), unless we are really convinced that it's the right thing to do. The theory provides a new view of gravity that resolves the old bugaboo of action at a distance, while not having to do so at the expense of positing communications of some sort limited to light speed. Not only this, but it also provides a counter motion to gravity that explains the recession of distant galaxies.

In fact, it's these two scalar motions that account for the general dynamics of the universe at large scales, and at small scales as well. The interaction of matter on very large scales produces a cycle, or evolution, that continues constantly and forever that relegates the effects of the 2nd law of thermodynamics to a relatively minor role in the universe, but it is driven by the dynamics of these two motions on a small scale.

What's fascinating about it is that so much of it is due to the limits of the system imposed by the discrete values of its sole component, space/time. At large scales, this limit manifests as what Larson calls a "gravitational limit." This limit is the point of equilibrium between the inward scalar motion of matter and the outward scalar motion of the universal expansion of space/time. It's an unstable point, because a slight increase in one or the other increases the effect in that direction; that is, there is positive feedback at the gravitational limit. This results in "clumping" or the aggregation of matter in regions of various volumes, and the dispersion of these volumes in general, while at the same time "locking them in" in relation to each other when the conditions are right.

On the other hand, at the opposite end, where matter is attracted to matter by gravity, another limit between these two motions is encountered, because of the mathematical nature of inverse relations. When atoms are able to approach one another close enough to encounter the limit of a single unit of space, the scalar "direction" of the gravitational motion, inherent in the scalar motion of the atom, is inverted, while the scalar "direction" of the universal motion is also inverted. Hence, the interaction between these two motions at that level is not unstable, but very stable, since a slight increase in one or the other decreases the effect in that direction and decreases it in the opposite direction; that is, there is negative feedback at this limit. This results in a bonding of the atoms involved, and, depending upon the dimensions involved in the interaction, the bonds may be one, two, or three-dimensional, producing the various phases of matter such as gas (0 dimensions), vapor (1 dimension), liquid (2 dimensions), or solid (3 dimensions).

Of course, the details are much more complicated than this, but that's the general idea. The complications, that are so hard to understand, both at the large scale as well as at the small scale, are due to the inverse, or reciprocal, relation of space and time in the equation of motion. Just as we've seen that numbers can have negative values without having to resort to "imaginary" numbers, because a true inverse is available when the relation between quantitative natural numbers, reciprocally related as ratios, is interpreted as a number in its own right, so too can physical entities have inverses, and for the same reason: the operational interpretation of magnitude is the same, quantitatively, on either side of the unit ratio.

So, a displacement of one unit on one side of unity is not any different than a displacement of one unit on the other side. The only way to tell a difference, is to assume a point of view, because of the perfect symmetry of the unit ratio. Clearly, if we say that s/t = 1/1, then s/t = 2/1 is just as plausible as s/t = 1/2. In fact, from the unit ratio point of view, they are the mirror image of one another, equal in all respects except in one: the one is the complement of the other. Indeed, they are actually two aspects of the same thing, just as the left and right ends of stick, or the left and right sides of a face: you can't have the stick or the face without their two, reciprocal, aspects; and you can't have the reciprocal aspects without the stick or the face. Like a horse, buggy and marriage, they all go together.

Likewise for scalar motion (motion not defined by a change in an object's location). Space and time are only its two, reciprocal, aspects. Without both of these aspects, there is no scalar motion. Without scalar motion, these two, reciprocal, aspects cannot exist. Hence, if we have one, we know we have the other. As we have seen in the previous posts above, the unit time displacement indicates a space motion of magnitude c that is stationary in space, the SUDR. Therefore, the unit space displacement indicates the inverse, or a time motion of magnitude c that is stationary in time, the TUDR. They are equal in magnitude, but they are the inverse, the mirror image, of one another.

Well, how can we deal with this idea of time motion? It's not going to be easy, that's for sure, but we've been dealing with it for decades now anyway. We just didn't know that it's what we were dealing with in relativity and quantum mechanics. It all started with the observations of Mercury's orbit, and the radiation of a black body, but that story will have to wait until next time. Meanwhile, feel free to ask questions.

Excal