In beginning the quantitative development of our RST-based theory, we incorporate the ancient principle of numbers in what we designate the operationally interpreted (OI) rational number, developed in the new reciprocal system of mathematics (RSM). In this view of number, the value of the number is the value of the relationship of two quantities. In the reciprocal system of physical theory (RST), this takes the form of two, reciprocal, changing, quantities, where the OI number represents the relative change rates of space and time in the equation of motion. Symbolically, this is written as
ds|dt = n|m,
where n and m are numbers corresponding to the reciprocal change rates of space, ds, and of time, dt, and the operational symbol employed to represent this relationship is the pipe symbol, “|”, rather than the division symbol, “/”, in order to distinguish the customary quantitative interpretation of the rational number from its new, operational, interpretation. The difference in the two interpretations is significant in the fact that zero is not an explicit part of the new system of numbers, just as it’s not an explicit part of the system of physical magnitudes in the RST; In other words, there is no such thing as zero motion, only a zero difference between two magnitudes of change, just as there is no such thing as a zero quantity, only a zero difference between two quantities.
This fundamental change in the concept of number, employed in the theoretical development of physical magnitudes, has a great effect on the normally accepted physical concepts such as space, time, and locations in space and time, which are used to define physical magnitudes in the legacy system of the mainstream scientific establishment. This is most easily recognized when the concepts of the Chart of Motion are understood, as these concepts extend the idea of motion, from an exclusive concept of the motion of objects, changing locations, in a container of space and time, to higher forms of motion, corresponding to higher forms of OI numbers. In effect, the properties of numbers and magnitudes are unified in the Chart of Motion.
The implications of this change are enormous, including its philosophical implications, which will be explored for centuries to come, if these developments prove to be as valid as they now appear. The essence of these philosophical implications is found in the difference between inventive and inductive science, which is profound. In short, the discovery of the concepts embodied in the Chart of Motion, the concepts of Larson’s reciprocal system, expressed in terms of a new system of numbers, enable the systematic development of physical theory, through the logic of deduction, from a few, fundamental, assumptions, induced within the limits of observation, rather than conjured up from the limitless imagination of human thought.
The observations upon which the assumptions of the RST are founded are the observations of expanding time and space. That the OI numerical representation of this universally observed expansion corresponds to a reciprocal relation of numbers, which constitutes the only observed relationship between space and time, places the RST, as a new system of physical theory, on incredibly solid, and unprecedented, philosophical grounds.
However, the current state of theoretical development reflects this philosophical strength only insofar as it constitutes an inflexible development of correctly deduced consequences of the fundamental assumptions. Of course, the certainty with which we can ascertain the validity of these theoretical deductions depends on the degree to which they correspond to observed, or measured, magnitudes of physical phenomena, which is never 100% accurate. Nevertheless, the comparison of theoretical entities, combinations of these, and relations between them, to corresponding observed entities, combination of these, and relations between them, are the elements that must comprise the science of RST-based research at the LRC.
The task is not an easy one. As John Baez has pointed out, “it takes guts to do physics” (see here.) However, we are quite gratified with the progress that we’ve been able to make thus far, which has come from following the logical consequences of the fundamental assumptions of the RST, even though this course has meant that the chain of our conclusions follows a different path than did Larson’s initial chain of conclusions. Happily, however, the determination of which course of development is the correct logical path to follow is testable, by comparison to physical observations. Obviously then, to perform these tests, requires more than the qualitative development of logical consequence. It requires the quantitative development of numerical magnitudes, with physical dimensions, that can be compared to observed physical magnitudes, with physical dimensions.
Nevertheless, the current system of measurements, and the accompanying concepts of physical dimensions, have been developed over centuries of work within the Newtonian system of physical theory, and, consequently, a translation between the legacy and reciprocal systems is required. One of Larson’s greatest contributions to science is his work in meeting this requirement. In his monumental Structure of the Physical Universe (SPU), in three volumes, he shows how the conventional cgs system of units, based on mass, length and seconds, can be expressed in the terms of the RST’s natural units of space and time. It should also be mentioned that, more recently, a theoretically unrelated, but similar, effort, with respect to the space and time dimensions of the international, SI, system of units, has been published independently by Engineer Xavier Borg (see here.)
In Larson’s theoretical development, he derives the natural units of motion from two, observed, constants of nature, the Rydberg constant and the speed of light constant. On this basis, the natural unit of space is calculated as
4.558816 x 10-6 centimeters, and the natural unit of time is calculated as
1.520655 x 10-16 seconds.
With these natural units, the natural units of acceleration, force, energy, etc, can be expressed in turn (see Chapter 13 of SPU, Volume II, Nothing But Motion). However, as the Chart of Motion shows, there is a distinction to be made between n-dimensional space|time units of M1, M2, M3, and M4 motion, which Larson did not explicitly recognize. Moreover, explicitly recognizing these different types of motion enables us to generalize the concept of number sufficient to conform to the concept of magnitude, unifying the two concepts in a most remarkable manner.
Armed with this additional knowledge of numbers and magnitudes, in the Chart of Motion, we should be able to develop the increasing degrees of freedom of fundamental numerical magnitudes to produce the fundamental entities of the RST-based theoretical development, in quantitative terms, that are related to, but different than, the corresponding entities in the LST-based theory, in certain cases. The most straight forward examples of this are seen in the cases of the concepts of velocity, energy, and momentum. They are very different concepts than those of LST physics.
For example, in RST-based theory, velocity is one-dimensional motion, or speed, with dimensions ds|dt, while energy is the inverse of this, or one-dimensional inverse speed, with dimensions dt|ds. Momentum, is two-dimensional energy, or inverse speed, with dimensions (dt)2|(ds)2, and is, therefore, a quantity of motion that is the difference between one-dimensional inverse speed and three-dimensional inverse speed. In the RST-based theory, matter is composed of units of three-dimensional speed, with dimensions (ds)3|(dt)3, while mass consist of inverse units of this motion, or three-dimensional inverse speed, with dimensions (dt)3|(ds)3. Meanwhile, two-dimensional speed, or inverse momentum, with dimensions (ds)2|(dt)2, is simply a quantity of motion that is the difference between one-dimensional speed and three-dimensional speed.
Clearly, this is a different system of units than that employed in LST physics. However, conformance to the LST system of units is not a requirement of nature. The only requirement is logical consistency. We know that, in the main, the LST system of units is logically consistent. Indeed, our modern technology is dramatic testimony of that fact. Nonetheless, it doesn’t follow from this that it is the only, or even the best, system of units conceivable.
The Chart of Motion reveals that both numbers and physical magnitudes correspond to geometric points, lines, planes, and volumes, or to 0, 1, 2, and 3-dimensional entities. The different types of motion in the chart, M1, M2, M3, and M4 motion, happen to correspond to the numbers 1, 2, 3, and 4, and the different characteristics of the four motion types stems from the different characteristics of the corresponding numbers, and the way motion can be defined using them.
What happens is that, as OI numbers, what we can term the “quantum degrees of freedom” of each of the three properties of the numbers, i.e. quantity, duality, and dimension, increase in ascending order. Thus, the number 1|1 has no quantum degree of freedom, but 2|2 has one quantity and one dual quantum degree of freedom (one in each dual “direction”), while 3|3 has two quantity and two dual, and 4|4 has three quantity and three dual quantum degrees of freedom. These four quantum degrees of freedom are shown below:
- 1|1 = 0 degrees of freedom
- 1|2, 2|2, 2|1 = 1 degree of freedom
- 1|3, 2|3, 3|3, 3|2, 3|1 = 2 degrees of freedom
- 1|4, 2|4, 3|4, 4|4, 4|3, 4|2, 4|1 = 3 degrees of freedom
In addition to the quantity and duality quantum degrees of freedom, shown above, there is also the dimensional quantum degree of freedom, and there are four of these, as well:
- (1|1)0, (1|1)1, (1|1)2, (1|1)3
- (1|2)0, (2|2)0, (2|1)0; (1|2)1, (2|2)1, (2|1)1; (1|2)2, (2|2)2, (2|1)2; (1|2)3, (2|2)3, (2|1)3
- (1|3)0, (2|3)0, (3|3)0, (3|2)0, (3|1)0; …(1|3)3, (2|3)3, (3|3)3, (3|2)3, (3|1)3
- (1|4)0, (2|4)0, (3|4)0, (4|4)0, (4|3)0, (4|2)0, (4|1)0; …(1|4)3, (2|4)3, (3|4)3, (4|4)3, (4|3)3, (4|2)3, (4|1)3
It’s not that there can’t be more than three quantities, or more than three dimensions, or more than three dual magnitudes. Obviously, there can be. However, Raul Bott proved that there is no new phenomena above three dimensions (see This Weeks Finds 105), and the perfect symmetry of 0-3 makes me suspect that the same thing holds for all three properties of numbers and magnitudes; that is, these are fundamental numbers and magnitudes, from which all other numbers and magnitudes can be constructed, but someone else will have to prove it.
In the meantime, I’m going to refer to this as the “fundamental magnitude conjecture” (FMC) in our RST-based theoretical development. The FMC is shown graphically in the four figures below:
Figure 1. FMC of Number 1
Figure 2. FMC of Number 2
Figure 3. FMC of Number 3
Figure 4. FMC of Number 4
In figure 2 above, the 3D magnitudes of number 2 are drawn in perspective. However, to avoid unnecessary complexity, the 3D magnitudes of numbers 3 and 4 are not drawn in perspective. Nevertheless, the 3D magnitudes of number 3 should be understood as a 4x4x4 cube, and the 3D magnitudes of number 4 should be understood as a 6x6x6 cube. Of course, the 6 unit line, the 6x6 unit square, and the 6x6x6 unit cube also contain the respective, n-dimensional, magnitudes of the 1, 2, and 3 numbers, as well.
The significance of the FMC is that all other magnitudes can be constructed from these fundamental magnitudes, but how they are constructed depends upon the type of motion employed. Beginning with M1, or unit motion, we see that this is not a zero magnitude of motion, but only a magnitude of motion without any degree of freedom. Without it, no other motion could exist. Then, by adding degrees of freedom to M1 motion, we make the other magnitudes of motion possible.
For example, adding one degree of quantum quantity to the ds|dt = 1|1 magnitude of motion, increases it to the ds|dt = 2|2 magnitude, which is equivalent to the 1|1 magnitude of motion, but the difference is that adding a degree of quantum duality won’t change the result of the 1|1 magnitude, but it will change the result of the 2|2 magnitude of motion. Similarly, adding a degree of quantum dimension to the 1|1 magnitude will not change the result of 1|1, but it will change the result of the 2|2 magnitude. Hence, we conclude that the 2|2 magnitude of motion is not the same as the 1|1 magnitude of motion, when the degrees of freedom of its properties are increased.
Nevertheless, the transition, from the traditional numerical magnitudes, based on the quantitative interpretation of number, to this fundamental concept of numerical magnitude, based on the operational interpretation of number, requires the recognition of the central change of concept: We are transitioning from the foundational concept that plays the central role in the Newtonian system of physics, to the foundational concept that plays the central role in the Larsonian system of physics.
That is to say, we are going from a static frame of reference, where the datum of physical and numerical magnitude is zero, to a dynamic frame of reference, where the datum of physical and numerical magnitude is not zero, but one. We still refer to zero magnitudes, but only as a difference, a delta, measure, not an absolute measure. The absolute magnitudes of RST-based theory do not include zero, as the four figures above show.
So, how do we translate between the two systems? Well, on the scalar level it’s trivial, but with increasing degrees of freedom, it’s a little more complicated, but not really a whole lot. What we have to recognize is that there is a difference between the direction of angle and the “direction” of duality. Hence, Increasing the dimensional degrees of freedom, increases the number of dual magnitudes, in non-unit magnitudes, as a power function.
For example, when the first quantity DoF is added to the 1|1 magnitude, to form the 2|2, it remains a point, indistinguishable from the 1|1 magnitude. Subsequently, adding the first duality DoF to the 2|2 magnitude creates the -1 and 1 magnitudes of the traditional, zero-based, integer system, except that the -1 magnitude of the new system is not an ad hoc invention, but actually the inverse of the positive 1 magnitude. Yet, when this is not properly recognized, only the positive magnitudes are treated as functions of dimensions. In other words, because the duality property of numbers is not recognized in the legacy system of numbers, the dimensional DoF is only applied to the positive magnitudes.
For example, in the binomial expansion, which is the dimensional expansion of the number 2 (equivalent to the number 3|1 in the new system), 20 = 1, but 21 = 2, not 4, and 22 = 4, not 16, and 23 = 8 not 64, as in the new system (see figure 3 above.) This is due to the fact that the duality property of number is not recognized in the legacy system.
What this means, then, is that the four numbers of the tetraktys, and the first four numbers of the binomial expansion, and the associated Clifford algebras of these numbers, and the legacy concepts of real, complex, quaternion, and octonion number systems, and the topological and physics concepts based upon them, are all missing a power of 2 worth of numerical magnitudes, which they would otherwise have, if the duality property (reciprocity) of numerical magnitudes were properly recognized.
For us, though, this amounts to recognizing that the scalar and pseudoscalar of the octonion, where (2/1)0 = 1 is the scalar point, and where (2/1)3 = 8 is the pseudoscalar volume, which contains the scalar point, represent only a part of the fundamental numerical magnitudes, in the universe of motion. There are actually the following magnitudes in the eight quadrants in the (3|3)3 graphic of figure 3:
- (3|1)x(3|1)x(3|1) = 23 = 8
- (1|3)x(3|1)x(3|1) = 23 = 8
- (1|3)x(1|3)x(3|1) = 23 = 8
- (1|3)x(3|1)x(1|3) = 23 = 8
- (3|1)x(3|1)x(1|3) = 23 = 8
- (3|1)x(1|3)x(1|3) = 23 = 8
- (3|1)x(1|3)x(3|1) = 23 = 8
- (1|3)x(1|3)x(1|3) = 23 = 8
Consequently, we actually have dual Charts of Motion, in our new system, each with four types of motion that can be identified as different modes of magnitude, which we will now designate as Ma, Mb, Mc, and Md, to ensure that we distinguish between these, as definitions, and the numbers of the chart themselves.
What we mean by mode of motion is the way a change of space and time is affected in the definition of motion. Of course, motion, by definition, is a change of space and time. In the first mode, Ma motion, the space|time change is the constant, continuous, all-directional change of the unit space|time expansion. In the second mode, Mb motion, time (space) expands continuously and uniformly, but the change in the space (time) aspect of this motion is the measured change of distance between two, or more, separate, objects. With one object at one location, and one or more objects at separate locations, this motion can be defined as a change of distance between the objects.
Hence, with no objects demarking spatial locations, or even just one object, demarking one location, this motion cannot be defined. With a minimum of two, separated, objects, however, the Mb motion between the respective locations is defined by either an increase in the distance, or a decrease in the distance between the objects, in a given time interval.
In the third mode, Mc motion, time (space) expands continuously, but the space (time) aspect of this motion is the measured change between three, or more, separate, objects. With one object at one location, and two or more objects at separate locations, this motion can be defined as a change of interval between a minimum of three objects, occupying three separate locations.
In the fourth mode, Md motion, both the space and time aspects of the motion expand, but only one aspect does so uniformly, while its reciprocal aspect continuously expands and contracts. This motion does not depend on the locations of objects for defining the required change.
With these four modes of motion, we can describe the magnitudes of the Chart of Motion in different ways. More on this later.