The First Postulate of Scalar Mathematics?
Arguably, the most profound consequence of the first fundamental postulate is that the theoretical universe is a universe of numbers, because, as I explained in the last post, all counting numbers can be represented as simple ratios. Another way of stating this is to assert that the set of counting numbers consists of one component, ratio, existing in three dimensions, in discrete units, and with two reciprocal aspects, numerator and denominator, an exact parallel to the first postulate of the RST.
But, the first objection to this assertion is that the usual mathematical definition of a rational number is that it is a fraction, or a quotient. For example, Wikipedia defines a rational number as "Any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number." Integers are discrete units, defined in Wikipedia as numbers "formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...)."
Figure 1. Wikipedia's Graph of the Integers and Their Negatives
Ok, but, according to this definition, since "...every integer is a rational number, [because in a/b] b may be equal to 1," then 1/1 and -1/1 (or, perhaps 1/-1), are integers. However, while this works well for the formulaic rules of algebra, any intuitive basis for understanding the underlying algebraic science is thrown out the window. In contrast, Larson realized that he could turn this around and redefine the integers, using rational numbers, so that 1/1 = 0, 1/2 = -1 and 2/1 = +1, restoring the intuitional basis of both integers and rational numbers. In this way, all integers, positive, negative and zero, are defined by the set of non-zero rational numbers, not the other way around. He called these discrete numbers "speed-displacements," and he derived them from the fundamental postulates of the system.
Unfortunately, he never regarded the first postulate as a mathematical proposition explicitly, but the parallel with the physical postulate is inescapable. Larson's objective was to develop a physical theory, and what he did to achieve his objective was nothing less than to redefine the age-old concept of space. He completely by-passed the ancient conundrum that continues as the central issue of physics today, where the contenders are absolutists and relativists, by redefining space as the reciprocal of time, in the equation of a universal motion, rather than a background set of positions satisfying the postulates of geometry. But in so doing he also redefined natural numbers as rationals, with zero as the unit rational, with multiples of 2/1 and their negatives as the reciprocal of these numbers, or multiples of 1/2. In other words, the graph of the number line in figure 1, above, is the numerical equivalent of PA 254 (see previous posts), as time and space march on, in the theoretical universe of motion.
This relentless progression is what Larson referred to as the "natural reference system." As it proceeds, there is a unit of time (positive integer) increase for every unit of space (negative integer) increase. That is, the universe of motion's space clock is the reciprocal of its time clock. Both quantities are increasing, but they are simply the reciprocal aspects of the postulated universal motion. As they continuously increase, the universe expands,and, at this stage, the universe is perfectly uniform, since nothing is perfect. For something to exist, Larson reasoned, there has to be a deviation from this perfect state of uniform motion.
The only option Larson could see that would constitute such a deviation from perfection was a displacement between space and time, or a change between the continuous increase of the denominator and the continuous increase of the numerator. Physically, the only way this can happen, he reasoned, is if, at a given space/time location in the progression, the next increase of one or the other of the two aspects of the motion decreases, instead of increases, and then continues to increase/decrease, repeatedly, thereafter. Since such an oscillation is a physical condition that is not ruled out by the fundamental postulates, its existence is possible and the consequences therefore have to be considered. In the case of the space aspect oscillating, this condition produces a new space/time ratio of s/t = 1/2, a -1 displacement. In the case of the time aspect oscillation, it produces a new space/time ratio of s/t = 2/1, a +1 displacement.
So, in the universe of motion, the number line in figure 1, the graph of the integers, not only represents PA 254, the uniform progression, but it also represents all the possible speed-displacements in the system, which are multiples of PA 252 (1/2) and PA 238 (2/1), as previously explained. It is these positive and negative units of speed, or motion, or at least their numerical counterparts, that we want to see if can combine into multiples. As the above Wikipedia article states, -1 is the additive inverse of 1, because adding them together gives us 0, the additive identity element. However, while this applies, when we are interested in the net displacement of a combination, it doesn't suffice when we want to know how to express the combined motion. For example, adding PA 252 (1/2) to PA 238 (2/1) does not give us PA 254 (1/1). worse yet, using Wikipedia's definition of rational numbers the answer is 2.5, which is not what we want at all. We could add numerators to numerators and denominators to denominators, but this is not really helpful, since 1/2 + 2/1 = 3/3 doesn't correspond to anything mathematically useful.
However, another possibility is to use the concept of the fulcrum and lever, a seesaw, if you will. We can write an equation for the weights on a scale like this,
L*l = R*l
where L and R are the left and right multiples of the unit weight and l = the distance the weight is placed away from the fulcrum. On this basis, we can write a similar equation for the RST's units of speed-displacement, where L and R are multiples of space and time speed-displacements and l is the "length" of time or space that the combination is "placed" away from unity:
s/t * t = t/s * s
With a single unit of speed-displacement on each side of the equation, at a "distance" of one unit of space and time duration, we get:
1s/2t * 1t = 1t/2s * 1s --> 1s/2 = 1t/2
Whether or not this equation can prove useful in the development of the consequences of the postulates remains to be seen, but one thing is clear: It's a valid mathematical consequence of the first postulate.
So, now, with first objection to the proposition that all is number satisfied by redefining the complete set of integers, as the discrete numbers generated by the set of non-zero rationals, which rationals are not the fractions of a whole, or quotients, but simply the difference between the numerator and the denominator, or displacement, the next objection to the proposition is that these numbers exist in three dimensions. But what is a 3D number?
Larson approached this challenge by inferring that the three dimensions of the first postulate are independent dimensions. However, this is essentially the same approach that the LST community uses, but then the motion in their system is not scalar, but vectorial, a 1D motion. They use the three axes of a 3D spatial coordinate system, the datum of which is zero, not 1. If we substitute Larson's 1/1 = 0, for the 0 in this system, we have a 3D system of motion, not a spatial coordinate system, and for that reason it's hard to see how we would use it. We can show the units of speed-displacements on each of the three axes, but what algebra do we use to combine them? Indeed, what meaning would they have, even if we could combine them in some way, since they are not positions that can be plotted as a set of curves?
In Larson's development, there would be two linear displacements along one of the axes to begin with, one displacement would be the LV+ and the other the LV-. Given an inward rotation of the outward LV+, the net displacement along the axis is supposed to be 0, but how can this be shown mathematically? Larson's rationale is that since the initial rotation is inward in 2D, it "kills" the outward progression in the remaining 2 dimensions left from the LV oscillation (the one dimension that the LV is propagating in, and the other "free" one.) So how do we show this on such a graph? If anybody has an idea, please let me know, because, if these are independent dimensions, then it's hard to see how to show that the linear and rotational displacements of his rationale add up to a net 0 magnitude, on such a graph.
In the meantime, we can see that, if the three dimensions are not independent, but scalar, then the initial LV expands and contracts along all three axes simultaneously. In the case of the space oscillation (LV-), 0D time continues forward normally, while 3D space oscillates, and in the case of the time oscillation (LV+), 0D space continues normally, while 3D time oscillates, giving us two, three-dimensional, functions, a space and a time function. The equation for the time function is
f(x) = s3/t0 * xt0
and the equation for the space function is
f(x) = t3/s0 * xs0
Since these 3D equations are written with the newly defined numbers that are a consequence of the fundamental postulates, the obvious question to be answered is can we regard them as legitimate 3D numbers, corresponding to the 3D speed-displacements of the RST? If so, can we combine them algebraically, using the fulcrum and lever equation? Finally, can we use these two 3D space and time functions to analyze the relationships of the various algebraic combinations of 3D numbers? In other words, does this number system constitute a mathematical basis for developing a theoretical universe of motion?
For me the answer is clearly yes, as iconoclastic as that conclusion may be.
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Does not make sense
Re: Does Not Make Sense
Hi Horace,
Good to hear from you again.
Thanks for the great question. Remember, the photon in my development of the RST is not an oscillating pseudoscalar. It is a combination of two oscillating pseudoscalars, one SUDR and one TUDR. The SUDR (space unit displacement ratio, s/t = 1/2) is a 3D oscillation of space, while the TUDR (time unit displacement ratio, s/t = 2/1) is a 3D oscillation of time. When the two are combined in a 1:1 ratio, the normal progression of time in the SUDR is the inverse of the normal progression of space in the TUDR. So, they have to progress at light-speed, but since they are joined at the hip, so-to-speak, they can only propagate in one direction, if they are to remain intact. The direction is defined by the three, orthogonal, dimensions of space, or time. The ultimate direction is, of course, random.
The difference of this approach viz-a-viz Larson's, is the assumption that the initial scalar expansion, which is 3D by the postulates, can, or cannot be reversed in one dimension only, while the other dimensions continue to increase normally. There are advantages to both approaches. I just want to see how far I can get with my approach.
You are an elusive man
Hi Doug,
You are an elusive man, you don't answer my emails and if I did not stumble upon this blog of yours I would forever watch the LRC or the rstheory.org forums or my inbox.
Anyway, I asked you these questions to refine your thinking. I hope you did not take it as hostile criticism of your thinking process, which you seem to endure a lot.
Although I do not negate its existence I have difficulty understanding how a SUDR and TUDR can be combined in a stable manner. I'd like you to elborate on this point of "hip-joining".
Doug wrote:
"...they can only propagate in one direction, if they are to remain intact. The direction is defined by the three, orthogonal, dimensions of space, or time. The ultimate direction is, of course, random..."
...but Larson wrote that any object, such as a photon, that has no capability of independent motion is stationary in natural reference system. Thus the direction of photon's propagation must be the fictitious result of relating it to the motion of gravitating observer (e.g. a material atom).
This leads us to a conceptual confict:
1) All normal matter moves the same way (creating mGRS, except for slow vectorial motions of our everyday lives)
a) All material atoms are all supposed to be a part of the same reference system (the mGRS),
2) All photons are all supposed to be a part of another kind of reference system (the NRS). See: http://www.reciprocalsystem.com/ce/refsys.htm
3) Photons are not supposed to have a capability of independent motion in the NRS.
Thus, the natural conclusion is that all photons should move scalarly away from material observers, yet the DO NOT !!!
Also, according to the 3 points above, a photon can never meet with another photon (but it apparantly happens in an interference) or with another atom (but it apparently happens in a reflection) !
Thus, it follows that either/or:
1) The direction of photon's propagation in mGRS is not determined solely by the relative motion of the material observer to the motion of the photon,
2) Photons apparently DO NOT move the same way because they can move toward or away (or at an angle) to matterial observers, albeit always away form material emitters.
3) Photon has a capability of independent motion in NRS (which you seem to suggest)
You seem to mean that photon's direction of propagation is random at the point of emission from an emitter (an atom), but how does it get memorized for later if we dare to propose that photon has a capability of independent motion in NRS ?
The same goes for the photon's polarizarion plane. If all atoms move in the same manner, creating the mGRS, yet we observe different polarizations of the photons, then the polarisations must be defined by the motions of the photons themselves- in other words the motion of the observing atom does not influence the polarization phenomena (nor the direction of its propagation). If we have many material observers, they will all see the same direction of propagation and polarization (except for enanglement, where two observers see the oposite polarization, and three observers see... ).
How does a non-3D phenomenon (such as polarization) arise only from 3D motions, and how does it get memorized if we dare to propose that photon has a capability of independent motion in NRS ?
Horace
P.S.
And why the hell is the photon sinusoidal ?
The superior performance of holographic notch filters proves that photon indeed is sinusoidal. See page 52, at:
http://books.google.pl/books?id=E9peWTnT9TgC&printsec=frontcover&dq=handbook+of+raman+spectroscopy&hl=en&cd=1#v=onepage&q&f=false
Bruce's explanation that the sinusoidal behavior is caused by the projection of counterspace turn onto normal space, is the only sensible answer to this question I have encountered so far.
Re: Elusive Man
Hi Horace,
Sorry, I've been looking for your emails. Maybe they got shunted off into another folder. I'll take a look tonight, when I get home.
Thanks for the questions. I welcome them.
First, you wrote:
The theoretical reason was just a guess, in the beginning, because they were inverses, but then we found physical evidence in oscillons. These are what they call the stable combinations of peaks and craters (3D inverses) in beds of small vibrating media. From the Wikipedia article, we read that the peaks repel peaks and craters repel craters, but they attract each other over small distances:
The fact that they then form these geometric patterns that the article refers to is very interesting, as well.
Second, you wrote:
Yeah, there is this big difference in the theory of the photons between Larson's development (RSt) and mine. In my development, the SUDR propagates in time, because all three dimensions of space (below unit speed) are vibrating, negating the forward progression of space, while the TUDR propagates in space, because all three dimension of time (above unit speed), are vibrating, negating the forward progression of time. Thus, it's the 3D reversals that bring the speeds of the SUDRs and TUDRs to zero-speed. When they combine together, however, they constitute a vibrating unit of space/time propagating outward from the source (point of combination perhaps) at c-speed, in a random direction.
Third, you wrote:
When Larson says the photons don't have independent motion, he means in terms of their propagation, but they actually have 142.222... degrees of freedom in terms of rotation. In my development, when a location of the progression is continually reversing spatially, it ceases to increase in space, because of the 3D space reversals. This means that the space progression is now expanding beyond it at c-speed, while it is carried along with the time progression. The inverse is true for a location continually reversing temporally. It ceases to increase in time, because of the 3D time reversals. This means that the time progression is now expanding beyond it at c-speed, while it is carried along with the space progression. From the perspective of either of these, we can construct a reference system of motion, but, at this point, we have two vibrating entities that are progressing orgthogonally relative to one another (one "in" space, and one "in" time), as shown in the graph below:
Figure 1. Space and Time Pseudoscalars Progressing Orthogonally
It's clear from figure 1, that a combination of these two oscillating entities will result in the combo's propagation along the diagonal; that is, the expansion of time in one, and the expansion of space in the other, produces unit motion in the combination.
Thus, in this model, any entity that is located at a point later in the time progression (i.e. all enties in our material sector), may be intercepted by the propagating combo, and any entity that is located at a point later in the space progression (i.e. all entities in the cosmic sector), may also be intercepted by the propagating combo.
Finally, you wrote:
Referring to the plot in figure 1, it's clear that a diagonal path increases in both space and time. All entities capable of independent motion in the material sector progress vertically (i.e. in time), while all entites capable of independent motion in the cosmic sector progress horizontally (i.e. in space).
Therefore, an entity that is not capable of independent motion progresses diagonally (that is, in space and time.) Hence, a combo (photon,) progressing (propagating) relative to the vertical and horizontal is capable of contacting entities coming up from below it and to its right (that is, previously existing in time in the material sector and at the correct spatial location), and it is capable of contacting entities coming from the left and above it (that is, previously existing in space in the cosmic sector and at the correct temporal location,) as time and space progress.
In other words, the propagating combo can collide with any stationary object that is at the right time and place. This would be obvious for waves expanding in 3D, but as the photon combo is constrained to 1 dimension by the bonding of the inverse entities, whether or not the collision takes place depends on its direction of emission.
As far as the "memorization" issue goes, it's clear that there cannot be a preferred orientation of the axis between the two oscillating pseudoscalars relative to matter aggregates, which mark different spatial or temporal locations relative to the emitter. The orientation is going to be random, but it is subject to change via the usual diffraction, refraction processes.
The polarization of the "spin angular momentum" of a photon is the result of the oscillation of its two, inverse, components (in the case of LST theory, this is rather vaguely, if not embarrasingly, referred to as the electric and magnetic field vectors, even though the transverse magnetic field vector is only implied by the changing electric field vector.) In the case of the SUDR|TUDR combo, we have two oscillating, inverse, magnitudes, and a variable phase relationship between them. If the phase difference is more than or less than 90 degrees, the photon can actually have "orbital angular momentum," as well, which is really interesting, since this is not restricted to "up" and "down" spin states, but an infinite number of orbital states (I don't know why they call them orbital - they should be called phase states, I think.)
In Larson's development, this is problematic, as there aren't two, vibrating, components in his model of the photon, but only one. In the new model that I'm working on, however, there are two components, the SUDR & TUDR, which can vary in spin up and down, as well as in relative phase, but exactly how they interact to produce the total angular momentum is still a matter under investigation. See here and here for some interesting aspects involved in the research.
In a P.S., you write:
Well, that is the question that launched a thousand ships, so-to-speak (not really, just two, I think - LOL.) Actually, it's not just a question of a sinusodial wave, which Larson's model can't answer to, but also the 720 degree spin cycle that neither Larson's nor the LST's model can answer.
Nehru's answer was his concept of bi-rotation and bi-direction, while my concept is the SUDR and TUDR combo's expansion/contraction. In both cases, one full cycle takes 720 degrees of rotation, accounting for the nature of quantum spin. The difference is that Nehru's model consists of two, quadrantal cycles (2 * (4 * 90) = 720), while mine consists of two bi-cycles (2 * (2 * 180) = 720).
Well, that's not the only difference, of course, but it's the difference in how quantum spin is accounted for in the two models. In the case of the expansion/contraction of my two inverse pseudoscalars, a 1 unit expansion/1 unit contraction includes the expansion/contraction of the diameters (1D), the spherical surfaces (2D) and the volumes (3D), corresponding to the electrical, magnetic and momentum (hopefully) properties of the photon. The first of these is sinusodial, when plotted, while the other two are sort of sinusodial (and very interesting to say the least.)
I hope this gives you something to go on, Horace. Great questions.
Doug