My apologies for not replying in so long. I haven't been able to keep up with the board, due to work scheduling. Hopefully next week, I'll have some more time to catch up and try to wrap my brain around this method.

__________________
"Reality is that which, when you stop believing in it, doesn't go away."
Philip K. Dick, Do Androids Dream of Electric Sheep?
"A lie can travel half way around the world while the truth is putting on its shoes."
Mark Twain

Avatar courtesy of Bunny.

Well, I'm back, after almost a year of being a away. I had to drop my contributions to this thread due to time constraints. A lot has happened in the past year, but Mike525 recently initiated a thread on the RST perspective of planet formation, which raised the subject of the RST in general, and offered me a chance to enter the dialog again.

However, to avoid hijacking his discussion, I thought it best to return to this thead, where we can discuss the RST in general. In his latest post to Mike525's thread, antoniseb replied to my post over there with the following request:

The above response was prompted in part, because, in answering an earlier request to "show how simple physics is expressed using Larson's ideas," I explained that it is important to recognize that Larson's theories are developed under a new system of physical theory, not Newton's system, where everything depends upon the function x(t) and its derivatives.

The difference is that, while the fields of modern physical science, from particle physics to cosmology, are based on the familiar concepts of the vectorial motion of objects, the new system is based on scalar concepts of motion defined without objects, and these ideas are new and unfamiliar.

However, while there is an important connection between these two concepts of motion, generally speaking the laws of physics, involving the motion of objects measured in terms of fixed spatial reference systems, and the familiar transformations of these, are not affected by the advent of the new system.

The domain of the new system is within the entities of radiation, matter, and energy, dealing with their inherent properties of mass, charge, spin, etc., which give rise to the effects of these properties, when scalar and vectorial motion between aggregates of matter exists, which sets the stage for the emerging concepts of force, acceleration, and momentum.

In the new scalar system, therefore, force is understood as a property of motion, not something that can exist autonomously, but this is not so in the vectorial system. The grand goal of Newton's program of research is to explain nature through the classification of a few elementary particles, by focusing on a few "fundamental" forces of interaction between them.

The mathematical and geometrical concepts used to clothe the concepts of vectorial motion, in a suitable formalism useful for conducting vectorial science, are naturally vectorial concepts, and since vectorial motion is the only motion recognized by the physicists and mathematicians that developed these mathematical and geometrical concepts, we tend to think that these concepts are all that exists, that the existing notions of mathematics and geometry, which pertain to vectorial motion, are concepts regarded as "the best of all possible worlds."

Well, Larson's works are changing all that, because the change of context that antoniseb inquires about takes us from a context in which the frame of space and time are paramount, into a new context where the frame of motion is paramount, motion defined in terms of space and time, but where they have no significance apart from their reciprocal relation in the equation of motion.

This means that the distance between the objects in a set of objects, which defines the concept of "space" in one, two, and three dimensions, is really only an emergent concept, the meaning of which is found solely in the history of the motion of the objects that separates them.

It's hard to overemphasize the impact that the redefinition of the traditional concept of space has on physics, and I'm sorely tempted to begin pointing out some of the more salient implications at this point, but I can't, if I want to limit the size of this post to any reasonable length.

If you read the previous posts in this thread, you will see that we were discussing Clifford algebras and Hestenes' Geometric Algebra (GA), and how Hestenes found that there is a surprising geometric interpretation of numbers that can be exploited to form an amazing algebra, not based on the use of imaginary numbers to form the one-dimensional complex numbers, or the two-dimensional quaternions, but rather based on the use of a new definition of vector operations, called the "geometric product," to form the three-dimensional octonions of the Cl3 Clifford algebra!

According to John Baez, the octonion numbers are the "crazy uncles kept in the attic" by traditional mathematicians (with some intriguing exceptions), because, though they are one of only four known normed division algebras, they are neither ordered, commutative, or associative, making them of little use to physicists, who have stuck with using complex numbers, the essential language of quantum physics.

However, Hestenes has shown that, with a geometric interpretation, these numbers form a set of four n-dimensional numbers, called blades, that can be used to great effect in the equations of theoretical physics, expressed as multivectors that contain one or more n-dimensional blades. The blades correspond to the four dimensions of the octonions (recall the binomial triangle at the fourth (2^3) line, explained in the previous posts of this thread), and constitute four grades of numbers:

• 0D number blades (2^0 = 1 grade, or scalar numbers)

• 1D number blades (2^1 = 2 grade, or vector numbers)

• 2D number blades (2^2 = 4 grade, or bivector numbers)

• 3D number blades (2^3 = 8 grade, or trivector numbers)

Therefore, an equation consisting of a multivector can contain all four of these blades in one! The utility of using this algebra is shown by Pezzaglia Jr, in his paper, Clifford Algebra Derivation of the Characteristic Hypersurfaces of Maxwell's Equations, where he combines all four of Maxwell's equations into one multivector, with each of the four blades yielding a fundamental law, or concept, of vectorial physics:

• The scalar blade yields Gauss's Law

• The vector blade yields Ampere's Law

• The bivector blade yields Faraday's Law

• The bivector blade yields the magnetic monopole

However, many mathematicians and physicists have trouble accepting the definition of the geometric product, because it combines the scalar of the inner product of vector multiplication with a special form of the outer product of vectors. How do you combine vectors with scalars?

Hestenes' GA shows how it can be done and in the process reveals the underlying connection between the numbers of algebra and the magnitudes of geometry, but it also reveals a fascinating drama in the evolution of mathematics as the language of physics.

In order to answer antoniseb's question, "Is there some way to apply Larson's concepts in a way that can be expressed mathematically?" I have to first set the context by delving into this mathematical drama, but I can promise you, if you will bear with me, it will be well worth the wait.

Excal

Thanks ExCal. I look forward to seeing the next installment.

__________________
Forming opinions as we speak

What Hestenes did was to take what is known as the operational interpretation of number, first recognized by Clifford, as opposed to the usual quantitative interpretation of number, and he used it to reinterpret the physical meaning of the imaginary number 'i'.

As I think we discussed last year, Michael Atiyah has characterized the imaginary number as the "biggest single invention of the human mind." He (and almost everyone else) is amazed by the fact that a pure invention, originally conceived to explain the negative roots of quadratic equations, ends up being essential to describe nature in quantum mechanics. It's just part of a great mystery as to how this concept could be essential in the appropriate language for physics, if mathematics is no more than a formalism, which can have little to do with a common underlying reality shared with physics.

Wigner referred to the "unreasonable effectiveness of mathematics in physics," as a "gift we neither understand nor deserve," but Hestenes sees the advent of the imaginary number as part of a deliberate attempt, lasting centuries, to generalize the concept of number to the point that algebra could be used to express the n-dimensional concepts of geometry.

In the final analysis, it is the operational interpretation of an imaginary number that makes Hestenes achievement possible. As he writes in his New Foundations for Classical Mechanics,

The way the two approaches are brought together in the geometric product is through the use of orthogonality. In the geometric product, orthogonality is expressed in the duality of the inner and outer vector product, which is equivalent to a certain rotation through an angle, as well as the orthogonal directions of a vector in a plane. In this interpretation, since i^2 is -1, which represents a 180 degree rotation through the complex plane from 1 to -1, then a 90 degree rotation is represented by i^1, a 270 degree rotation is represented by i^3, and i^4 brings us back to the beginning, or to i^0, on the unit circle.

However, GA does more than provide us with a neat package for unifying the concepts of the vectors and spinors in one language. As impressive an achievement as this is, it is crucial to recognize that, in the process of giving geometric meaning to the ad hoc invention of 'i', GA also unifies the Clifford algebra, Cl3, with the concepts of vectorial motion and Euclidean geometry, in that it permits numbers to represent n-dimensional magnitudes with direction.

This long-sought generalization of number unifies the binomial expansion of n-dimensional numbers, the associated Clifford algebras, and the elements of Euclidean geometry, when it is realized that the expansion of the numbers in the associated algebra are isomorphic to the expansion of a zero-dimensional point into a one-dimensional line, a one-dimensional line into a two-dimensional plane, and a two-dimensional plane into a three-dimensional cube.

As the expansion grows, from 0 to 3 dimensions, the number of elements in the algebra grows as the number of elements in a cube; that is, at 2^0 = 1, we have only the concept of points in the geometry and only scalar numbers in the algebra, but at 2^1 = 2, we have both the concept of points and lines in the geometry, and scalar and vector numbers in the algebra. At 2^2 = 4, we have the point concept, two line concepts (orthogonal lines), and the concept of the plane in geometry, and scalar, vector, and bivector numbers in the algebra.

Finally, at the 2^3 = 8 dimensions, we find the culmination of all these in the maximum dimensions of the geometric cube: There are the concepts of points, lines, planes, and cubes in 3D geometry, and the corresponding concepts of scalar, vector, bivector, and trivector numbers in the 3D algebra. Moreover, thanks to the operational interpretation of number and Hestenes work, the scalar, vector, bivector, and trivector numbers of the algebra can also be thought of as scalar magnitudes, linear magnitudes, quadratic magnitudes, and octonic magnitudes, generated by rotation, i.e. vectorial motion.

Thus, Hestenes has unified the concepts of vectorial motion, mathematics, and geometry, in his GA. However, the significance of this achievement is not just academic, but has huge ramifications in physics that I don't believe the mainstream physics community fully appreciates, at this point in time, preoccupied as they are with the string theory controversy, and the fundamental crisis in theoretical physics that gave rise to string theory, but which it seems only to be able to exacerbate.

Nevertheless, the mathematical and geometrical concepts clarified by GA, illuminate the central problem at the heart of the current crisis in theoretical physics: the issue of reconciling the concept of the continuum, upon which the fields of the general relativity theory of gravity are based, and the concept of the quantum, upon which the fields of the standard model theory of particle physics are based. The essential clue that GA provides is that while there is a fundamental disconnect (no pun intended) between the two concepts, they both suffer from the same ailment: the dreaded singularities, or infinities caused by the numerical/physical concept of zero.

To see how GA relates to the problem of infinities requires us to now turn our attention from the subject of mathematics to the subject of physics. In the previous posts of this thread last year, we discussed the ideas in Lee Smolin's paper on background independence, where he identifies five major, unsolved, problems in theoretical physics, and how these are related to the famous debate between Newton and Leibnitz over the nature of space, which is reincarnated in the argument for the background dependence of quantum physics.

Since then, Smolin has published a book entitled, The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. In the first part of the book, he again reviews the five unsolved problems of theoretical physics and in this connection writes:

referring to the revolution of the last century that started with the discovery of the quantum. This revolution is unfinished, according to Smolin, in contrast to the previous revolution that started when the continuum secret of the Pythagoreans' square root of 2 got out in the open, and finally culminated with Newton's overthrow of the Aristotelian theories of space, time, motion and cosmology that competed with the ideas of the Copernican revolution.

The challenge of reconciling these two concepts, one discrete and the other continuous, did not seem unsurmountable in the early days of the unifinished revolution, but today it is more intractable than ever. The problem is not that we don't have theories that work, but that the theories we have work too well, yet are incompatible. Smolin writes:

However, though this is the reason most cited for why we need a unified theory, it is not the only reason. There is more to it than that, and it has to do with the infinities that plague both the continuum based theory of gravity and the quantum based theory of particle physics. Regarding the former, Smolin writes:

Of course, this has lead to the effort to find a theory of quantum gravity, but the effort has been unsuccessful to date, and though quantum physics dealt with its own problems of infinities long ago, through the suspicious procedure of "renormalization," that has effectively swept them under the rug, it is hoped that a unified theory can be found that will include gravity and, at the same time, tame the infinities of quantum theory without recourse to renormalization. Smolin continues:

Clearly, the problem of infinities is related to the continuum. Given Zeno's paradox, there is no limit to how finely a continuum can be subdivided, whether it's the continuum constituting 1D linear space, or it's the continuum of a 3D volume of space. The reason string theory has been developed and highly touted is precisely because the nature of a string excludes the concept of zero, which corresponds to the concept of a point.

If particle physics can be described without recourse to the concept of point particles, or the infinite variables of a field, and the warping of a spacetime continuum, required to describe gravity, can be accomplished by the same means, then that way out of the impasse seems very promising.

However, the problem with the string theory approach brings us back again to the nature of space and time, but, in this case, the plague of the infinities of the continuum attack again, albeit from a different angle: string theory requires 10 dimensions of space, but there are only four dimensions of spacetime in current theory, leaving six dimensions that have to come from somewhere else. The thought is that these extra dimensions could be "compactified," but there are essentially an infinite number of ways to compactify them!

Thus, we have a fundamental crisis in theoretical physics on our hands that is being caused by the fact that the dual of the point is infinity, and though nature knows a way to deal with this fact, evidently, we don't.

In the meantime, Hestenes has found a way to unify the concepts of vectorial motion, algebra, and geometry. Could it be that this is a prerequisite to unifying the laws of physics? We'll discuss it next time.

Just as a reminder, this is the third post, which constitutes the latest part of my answer to antoniseb's question:

In the first post, I explained that the scalar system is different than the vector system and that this difference is reflected in the mathematical and geometrical concepts developed to apply the system in a program of scientific research. In the vectorial system, motion is measured in terms of an object's change in locations, and locations are separated by distances in terms of magnitude and direction, the "space" between objects.

However, we see that the RST redefines space, as simply the reciprocal quantity of time in the equation of motion. Consequently, with this redefinition of space, the "space" between objects has no independent meaning; that is, it is emergent. Nevertheless, it is this "space," defined by a set of points, that satisfies the postulates of geometry, and forms the basis for vectorial motion. Newton referred to this, when he wrote that geometry has nothing to say about how "right lines and circles" are drawn. These matters are outside the domain of geometry, and must come from mechanics, or the vectorial motion of objects.

So, we turned to the subject of how mathematics has developed historically, as a language of physics, in terms of the effort to generalize the concept of number from a limited concept of counting, to a more general concept of magnitude, the magnitudes of geometry, which magnitudes are points, lines, areas, and volumes.

In the second post, we considered how numbers were provided with the geometric property of direction, by means of the ad hoc invention of imaginary numbers, and how this approach was considerably improved by Hestenes, who was able to show that, by exploiting the operational interpretation of number, it's possible to define a direction property of numbers, without recourse to the ad hoc invention of imaginary numbers.

We discussed how the consequences of Hestenes' achievement are deeply significant, that his achievement constitutes more than an efficient method for writing equations, that it actually unifies the concepts of vectorial motion, algebra, and geometry in three dimensions for the first time. Then, we discussed the conclusion that this unification of the concepts of motion, numbers and geometry goes right to the heart of the modern crisis in theoretical physics, the challenge of reconciling the dual nature of the physical structure of the universe, the nature of the continuum and the nature of the quantum.

The basis of this conclusion is what we will discuss in this post. It has to do with the infinities that plague current theory, and the observation that, while these infinities are obviously the property of the continuum, the infinite values of magnitudes in the continuum are necessarily associated with the concepts of vectorial motion. Of course, the implication is that, with the advent of scalar motion, the seemingly insurmountable problem with infinities in the equations of motion, automatically disappears.

When I first discovered Hestenes' GA, in the context of my study of scalar motion, I thought it might prove to be the basis for a new mathematical language useful for expressing Larson's scalar motion concepts mathematically. Unfortunately, this was not the case. Ironically, however, it was GA's clarification of the concepts of vectorial motion that actually enabled me to discover that there exists a scalar concept of n-dimensional numbers that enables us to express the concepts of scalar motion mathematically.

In discussing this discovery, it's important to recognize that the insights involved came to me in fits and starts, and many times they were not clear until viewed in retrospect, after pressing ahead with nothing but the assumptions of the RST to guide me. Thus, this is the newly minted coin of a new territory we will be discussing here. It is consistent with Larson's new system, but was completely unknown to Larson, since it was only recently developed, more than a decade after his decease.

The key that opened the door to n-dimensional scalar mathematics, which I call the Reciprocal System of Mathematics (RSM), for obvious reasons, was the same key that unlocked the door for Hestenes, enabling him to formulate the n-dimensional numbers, or k-blades, of his algebra, the combinations of which constitute the multivectors of GA. That key is the operational interpretation (OI) of number.

Hestenes used it to define the geometric product in terms of the inner and outer products of vectors. The geometric product is an amazing innovation, because it defines a numerical value in terms of the relation between the directions of vectors, and thus it makes it possible to construct an algebra of these numbers, which doesn't suffer from the problems of numbers constructed with the traditional combinations of real and imaginary numbers, the complex and quaternion numbers.

The significance of this new algebra, then, is that it is literally a three-dimensional algebra, whereas the algebra of complex numbers is one-dimensional, and the algebra of quaternions is two-dimensional, but the significance of this fact cannot be fully appreciated, until we recall that vectorial motion is always one-dimensional.

This fact sets the stage for the drama we are experiencing in mathematics and physics today, because, in order for us to conceive of one-dimensional motion, three-dimensional algebra, and three-dimensional geometry, coexisting in some notion of a unified structure, the n-dimensional numbers of the algebra and the n-dimensional "space" of the geometry, have to contain the one-dimensional motion.

When one-dimensional motion is contained by n-dimensional "spaces," it is possible to define the vectorial motion of objects directly with three-dimensional algebra. In contrast, the one-dimensional algebra of complex numbers cannot define vectorial motion directly, it can only define it indirectly, by defining the points that are necessary to describe the historical path of vectorial motion.

Thus, using the algebra of complex numbers, we can define the function x(t), as a set of points in the complex plane, and we can define the differentials and integrals associated with the curve of the function, plotted with the numbers of the algebra. In other words, the one-dimensional complex algebra exploits the infinite points of the continuum to enable the calculus, which is the foundation of the modern world's science and technology.

We can do the same thing with GA, but without having to explicitly define the points of the curve, because GA's multivector can represent rotation directly, in the form of a bivector. Nevertheless, doing this is sort of like using a wheel to measure distance; it's a great way to measure length in some ways, but it hardly characterizes the potential contribution of the wheel to technology.

Our ability to exploit the extended power of GA will be limited until we gain greater insight into its significance, but the fact remains, it is literally two orders of magnitude more advanced than the algebra of complex numbers. In its one-blades, we have 1D numbers equivalent to magnitudes of linear motion, with two directions (positive and negative). In its two-blades, we have 2D numbers equivalent to magnitudes of rotational motion, with two directions (clockwise and counterclockwise), and in its three-blades, we have 3D numbers equivalent to magnitudes of scalar motion, with two directions (in and out).

However, currently, as far as I know, the magnitudes of the bi-directional scalar motion of the three-blades are not recognized for what they are. The 3-blade is labeled a trivector, and its space magnitude is viewed as a volume, but the concept of this number's equivalence to an outward/inward magnitude of motion is not generally discussed. Clearly, however, GA's linear 1-blade numbers are capable of expressing 1D magnitudes of simple harmonic motion (SHM), in one-dimensional equations of motion. Likewise, its linear 2-blade numbers are capable of expressing 2D magnitudes of torsional SHM, or rotational vibration, in one-dimensional equations. Consequently, it's just one more small step for us to conclude that its 3-blade numbers are capable of expressing 3D magnitudes of SHM, in one-dimensional equations, though I don't know if there is a name for this motion per se.

Nevertheless, if we describe these three magnitudes in terms of directions, the 1-blade number is equivalent to magnitude in one specific direction within the three dimensions of a volume, the 2-blade number is equivalent to magnitude in all the directions of one specific plane, within the three dimensions of a volume, and the 3-blade number is equivalent to magnitude in all the directions, within the three dimensions of a volume. Hence, when the GA magnitudes are applied to the description of vectorial motion, the 1-blade motion is the familiar translation of an object, the 2-blade motion is the familiar rotation of an object, and the 3-blade motion is the familiar expansion of an object.

Of course, SHM is alternating motion in the two "directions" of each blade, where, by placing the word "directions" in quotes, we are referring to the duality of each blade; that is, the negative and positive "directions" of the 1-blade, the clockwise and counterclockwise "directions" of the 2-blade, and the outward and inward "directions" of the 3-blade, constitute the duality property of each blade.

Clearly, as the dimensions of the k-blades in GA is three, there are three independent 1-blade numbers (three 1D numbers with orthogonal directions, or three vectors), three independent 2-blade numbers (three 2D numbers with orthogonal directions, or three bivectors), but only one 0-blade number (one 0D number with no directions, or scalar), and one 3-blade number (one 3D number with all directions, or pseudoscalar).

Of course, the members in each set of numbers, or blades, follow from the inherent duality of the numbers characterized by the grades: Since the duality of the scalar, or 0-blade, has no direction, it is always characterized by 2^0 = 1 duality, while the positive/negative duality of 1-blades is two, so it is always characterized by 2^1 = 2 duality, and the duality of the 2-blade is four, because it is composed of two 1-blades, so it is always characterized by 2^2 = 4 duality, and the duality of the 3-blade is eight, because it is composed of three 1-blades, so it is always characterized by 2^3 = 8 duality.

Hence, it's this dimensional expansion of duality that gives GA its power, in effect, defining 1 + 3 + 3 + 1 = 8, mathematical dimensions, within the 0, 1, 2, and 3 dimensions of geometry. If the geometric dimensions are counted, there are four of them, and then their sum is 1 + 2 + 3 + 4 = 10, which is the sum of the 3D binomial expansion, and the spatial dimensions of string theory. Some think that this is just coincidental, but perhaps not. If it is not, then could the compaction of these eight mathematical dimensions into three geometric dimensions, be another, more successful approach to string theory? I believe it is, but, regardless, that fact is just a distraction at this point, because our goal is to get to a new scalar science, through a new definition of space, as the reciprocal of time. String theory is a concept of vectorial science, and therefore we don't want to go there, but it sure is tempting (maybe in another life!)

Nevertheless, this neat packaging of algebraic numbers, with geometric magnitudes, exhibiting an almost breathtaking symmetry of duality and revealing a deep and mystifying connection of mathematics, geometry and vectorial motion, suggests a common underlying reality that has always been suspected, but only teasingly revealed in modern science.

Could it be that the scalar motion concepts of Larson are also part of this new-found unity? Is it possible that discrete values of GA's n-dimensional numbers could be used to represent discrete values of n-dimensional magnitudes of scalar motion? Although Larson never heard of GA, and probably knew nothing of Clifford algebras and the symmetry of the expansion of duality, the binomial expansion, his development of the world's first general theory of the universe, using his scalar system of physical theory, came very close to answering this question, breathtakingly close.

(to be continued)

Larson proceeded to develop his scalar motion theory from the basic postulates of the RST, without the benefit of Hestenes' work. Therefore, the concepts of duality and dimension, and how they are related to each other, as revealed in the binomial expansion, and the concepts of Clifford algebra that encapsulate these ideas in terms of n-dimensional numbers, though they were certainly available to informed specialists in the field of mathematics, during Larson's time, they were scarcely visible to many uninformed specialists, let alone non-specialists. They didn't become generally known until Hestenes' efforts to use them as the basis for a new, unified, language of physics, began to gather steam in the eighties and nineties.

Today, we know that the purpose of GA, according to Hestenes, is to facilitate physics educational and research efforts. In delivering his 2002 Oersted Medal Lecture, "Reforming the Mathematical Language of Physics," Hestenes told the audience:

Then he listed five features of the new language:

1. GA seamlessly integrates the properties of vectors and complex numbers to enable a completely coordinate-free treatment of 2D physics.
   
2. GA articulates seamlessly with standard vector algebra to enable easy contact with standard literature and mathematical methods.
   
3. GA Reduces "grad, div, curl and all that" to a single vector derivative that, among other things, combines the standard set of four Maxwell equations into a single equation and provides new methods to solve it.
   
4. The GA formulation of spinors facilitates the treatment of rotations and rotational dynamics in both classical and quantum mechanics without coordinates or matrices.
   
5. GA provides fresh insights into the geometric structure of quantum mechanics with implications for its physical interpretation. All of this generalizes smoothly to a completely coordinate-free language for spacetime physics and general relativity to be introduced in subsequent papers.

However, what few people knew, or could have even cared about at the time, was how the language that simplifies and clarifies the structure of vectorial physics, would also illuminate the work of Larson eventually. Indeed, its contribution to the science of scalar physics, may eventually prove to be more valuable to science in general than its contribution to the science of vector physics, since physicists and educators, well trained in the traditional language of vector and complex algebra, are relunctant to replace the known and familiar language of physics, with an unknown and unfamiliar one.

It may seem ironic that a new system of vector algebra, designed to improve the existing mathematical language of the mature science of vector physics, ends up providing a new mathematical language for the immature science of scalar physics. Yet, it is easy to understand it, when one looks at GA, not as a geometric algebra, but as a multi-dimensional number system that contains, as an integrated whole, advanced concepts of number that correspond to both scalar and vector magnitudes.

In starting our discussion of this expanded view of GA, it's important to recognize that there is a pleasing, but mysterious, symmetry in the grouping of its n-dimensional basis numbers: 1 + 3 + 3 + 1 = 8, which seem to pair vectors and bivectors (3+3), and scalars and trivectors (1+1). When we interpret this mathematical symmetry geometrically, as the 2x2x2 cube, with the scalar basis point at the intersection of the three orthogonal basis vectors and the three orthogonal basis bivectors, a point which coincides with the center of the eight corners of the pseudoscalar basis trivector, the physical beauty in the symmetry of its form, the conceptual beauty in the symmetry of its algebra, and the intellectual beauty in its complete and flawless union of the meanings of direction, duality, and dimension, is simply startling in its power to make one marvel.



Figure 1. Larson's Cube

I've dubbed it "Larson's Cube," because he initially used it to illustrate the eight units of three-dimensional scalar motion in the RST, but, now, when I look at it, it's as if I see the physical, mathematical, and intellectual structure of the entire universe contained in its simple, three-dimensional, figure. It seems to me to contain the great secret of nature's unity, as it were, the secret to how she unifies the continuum and the quantum, the secret that is the Holy Grail of theoretical physics.

Of course, I can't prove these things yet, but I'm so convinced of their veracity that I have organized and incorporated the "Dewey B. Larson Memorial Research Center," or "Larson Research Center," for short, to employ the mathematicians, physicists, and other professionals who can perform the scientific work necessary to prove it, to the extent that it may be possible to do so.

The reason that I see so much beauty in this cube is that it seems to contain every element that is necessary for a theory of everything, but nothing more. It integrates the concepts of motion, mathematics, and geometry into one graphic, which can be viewed as a single expression of that motion, which in the RST, we assume is the one component of which the universe consists. To see it expressed in one graphic, is tantamount to looking at the very foundation of the universe, in all its awesome power.

Clearly, however, I'm getting ahead of myself. To understand the power of Larson's Cube, we have to understand how it integrates the science of vectorial motion, with the science of scalar motion, and it all begins with the power of one, but not the quantitative interpretation of the number one, but the operational interpretation (OI) of the rational numbers as one.

As the ancient Greeks understood it, the number one (monad) represents eternity, and they probably learned it from the Egyptians, who in turn probably learned it from the Hebrews. In our OI number one, 1/1, we can understand this as signifying infinity/infinity is equal to 1/1; in other words, the number one contains all the numbers, 1/1, 2/2, 3/3, ...n/n, where n goes to infinity.

Normally, this fact is regarded as trivial, until, that is, we realize that this unity, apeiron, to the ancient Greeks, is a principle of reality that subsumes (i.e. contains, encompasses) peiron, or that which is limited, bounded, finite. These are the notions of continuum and quantum, the very notions that are plaguing theoretical physics. Larson's RST necessarily begins with this unity, which he calls the "unit progression" of space and time, as the initial condition of the universe of motion, because he defines space as the reciprocal of time, as motion, and motion as the sole component of the universe.

However, in describing the ancient Greek idea of this, we read in Wikipedia:

Larson never spoke of ancient Greek philosophy in this connection, as far as I know, but came to his conclusions based on his work to determine the inter-atomic distances in elements, in which he found that, only if he assumed a reciprocal relation between space and time, would his calculations work out:

However, Larson's initial reaction to this thought was negative, probably because he was thinking of space in terms of the distance that separates objects:

Intrigued, he pursued the idea:

In just this manner, his work began to bear fruit. One of the first things Larson had to do was to understand how, given this idea of the unit value in a universal motion in which space progressed as the reciprocal of time, deviations from this unit value could occur. In other words, he needed to know how n units of space, or n units of time, could come to be associated with one unit of its reciprocal aspect, so that a deviation from the unit, 1/1, condition occurred. Without such a deviation, where ds/dt = 1/n or n/1, the universe of motion is a great void, the perfection of nothing.

According to the Wikipedia article, this same challence faced the ancients:

Again, however, while Larson’s approach is much less philosophical, it amounts to the same principle in the end:

Hence, just as the ancients insisted, the play of apeiron and peiron commences according to principles of harmonia.

(to be continued)

Larson gave us very few diagrams to illustrate the principles of scalar motion, but in the previous posts, I introduced an algorithm, called the progression algorithm (PA), that produces a visual output that is very helpful in studying the scalar space/time progression ratio. The unit progression PA is



Figure 1. Unit Space/Time Progression PA

The unit progression is the datum, the point of reference for the scalar motion of the universe of motion, the apeiron, the perfect symmetry of the Reciprocal System's initial state, the space/time ratio equal to a 1/1 space/time progression. We call it the natural reference system of motion, the physical "zero" of the system, from which all physical activity is measured in the RST, and we use the PA in figure 1 above to generate this state of scalar motion as the unit progression ratio of space/time.

In the previous posts of this thread, I've explained how the symmetry of this 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." The PAs in figures 2 and 3 below generate these two progression ratios.



Figure 2. The Unit Time Displacement PA



Figure 3. The Unit Space Displacement PA

These two units of magnitude exist on either side of the unit ratio, and, taken together, the three of them correspond exactly to the initial three numbers of the integer number system, generated by the operational interpretation of the magnitude of rational numbers. In fact, we can easily summarize all this information with just a few numbers. The three numbers:

1/2, 1/1, 2/1,

which, operationally interpreted, are equivalent to three integers

-1, 0, +1,

because the change in the rates of progression of space and time, caused by the "direction" reversals in one aspect or the other, confines the progression of the oscillating aspect of the progression to be confined to one unit. In scalar physics, then we have a process, "direction" reversals, that, given a unit space/time progression, ds/dt = 1/1, producees two new ratios, ds/dt = 1/2 and ds/dt = 2/1, which constitute discrete, unit, magnitudes of oscillations, or motion, the peiron, emerging from the apeiron.

Now, one of the first things we want to know is, "What can we do with these numbers?" We want to know if we can add, subtract, multiply, divide, raise their powers, and extract their roots, because, if we can, then we have an algebra of scalar motion, and it will go a long way towards helping us develop a scalar science around the scalar system.

Larson was asked, many times, if there were a new mathematical formalism to go with his new system, and he insisted that there wasn't, because there wasn't any need for one, that the primary contribution of the RST is a clarification of the concepts of motion, not an addition to the already vast field of mathematics.

However, we now see the mathematics of a new system of numbers emerging from the mathematical equivalent of the RST assumptions. Whereas, in the RST, we assume that one component, motion, with two reciprocal aspects, space and time, is the initial condition of the physical universe, in the RSM, we assume that one number, an operationally interpreted ratio, with two reciprocal aspects, numerator and denominator, is the initial number of the mathematical universe.

Given this operational interpretation of the ratio of integers, as a signed integer itself, several things happen. First, we remove the necessity of qualifying zero as a number, which of course, it isn't. Zero is an important concept in its own right, to be sure, but it isn't a number. At least now it isn't. Second, we have no need to use signs with OI numbers, where the number itself is unique, and unambiguous. Third, we have no concept of orthogonality. The dimensions of OI numbers cannot include the notion of multiplication in the sense of magnitudes that are the product of two, orthogonal magnitudes of direction, because OI numbers are scalar numbers, and scalar numbers do not have direction, or othogonality.

However, given their equivalence to integers, we already know a lot about how they constitute a mathematical field and all, because they are isomorphic to the integers; that is,

1/n, ..., 1/2, 1/1, 2/1, ..., n/1,

is isomorphic to

-x, ..., -1, 0, 1, ..., x,

so anything we know about the integers, holds for the OI numbers, as well.

Therefore, if we add 1/2 = -1, to 2/4 = -2, we get 3/6 = -3, and, in general, m/n + m/n = 2m/2n, and so on.

Nevertheless, there is a big difference when we add equal OI numbers on opposite sides of unity, because not only are they equal in magnitude, but they are inverses as well. Therefore, like equal weights on the opposite sides of a pan balance, they balance each other, they do not eliminate one another. For example, a one pound weight on one side of a balance exactly offsets an equal weight on the opposite side, but together they weigh two pounds. In contrast, with integer numbers, a negative number cancels an equal positive number, requiring the concept of zero, the bane of modern theoretical physics.

Hence, we when we add 1/2 + 2/1, we get 3/3 = 1/1, not zero, but the operation is isomorphic to -1 + 1 = 0. It's just a different animal. Now, since n/n = 1/1, then

2/2 + 2/2 = 4/4 = 1/1,

a balanced OI number. Nevertheless, in the case of the ds/dt = 1/2 case of the time displaced PA, for every two units of time progression there are still two units of space progression, the difference being that one of these two space units is an increase (arrow points to the left), and one is a decrease (arrow points to the right). Thus, if we add the respective numbers of the two unit displaced PAs, we get

(1/2 + 2/1) = 3/3,

because we have not counted the decreasing space unit (arrow to the right) on the time displaced PA, nor the decreasing time unit (arrow to the left) on the space displaced PA. Adding a term to account for these units, gives us

(1/2 + 1/1 + 2/1) = 4/4 nm,

where "nm" is the unit of "natural motion." This equation is a fundamental equation of the RSM, and is called the first, reciprocal, number (RN) of the RSM. Again, given that we are assuming that the OI number system is isomorphic to the integer number system, the RN, because it is a sum of OI numbers, should also be an integer, and it is, though 4/4 = 1/1.

However, if we square (1/1), we get (1/1), but if we square (4/4), we get (16/16), and if we cube (4/4), we get (64/64), and even though 4/4 = 16/16 = 64/64 = 1/1, 4/4 units of motion squared is

(1/2 + 1/1 + 2/4)² = (4/8 + 4/4 + 8/4) = 16/16 nm,

and

(1/2 + 1/1 + 2/4)³ = (16/32 + 16/16 + 32/16) = 64/64 nm

a completely different result.

There's much more to say about OI numbers, but I'll have to wait until next time to continue the discussion, including how we get the important results above.

Since reciprocal numbers (RNs) are composed of OI numbers, and OI numbers are isomorphic to integers, they do not have the property of direction. However, they do have the property of "direction," because just as an increasing negative integer increases in the opposite "direction" with respect to increasing positive integers, an increasing OI number on one side of unity increases in the opposite direction with respect to increasing OI numbers on the other side of unity. Thus, 1/2 + 1/2 = 2/4 = -2 is "farther away" from unity than 1/2 = -1, but in the opposite "direction" with respect to unity than a similar increase on the positive side of unity would be.

However, given two RNs such as, for example,

(1/2 + 1/1 + 2/1) = 4/4 and (2/4 + 2/2 + 4/2) = 8/8,

there is no orthogonal relationship of directions that can be defined when combining them, just as there is no orthogonal relationship of directions that is definable when physical scalars such as points, or pseudoscalars such as volumes, are combined. Consequently, when multiplying RNs, though we multiply each term in the multiplicand by each term in the multiplier, as we would if they were vector magnitudes, the product "n times m" literally means that n is repeated m times, not that the scalar unit is changed to an "n by m" area unit by the operation. Also because the numbers are OI numbers, not fractions, we multiply numerators by numerators and denominators by denominators in each term, so the product of the two RNs is

(1/2 + 1/1 + 2/1) * (2/4 + 2/2 + 4/2) =
(4/4) * (8/8) = 32/32 nm,

because multiplying three terms by three terms,

(1/2 + 1/1 + 2/1)
(2/4 + 2/2 + 4/2) =

2/8 + 2/4 + 4/4 + 2/4 + 2/2 + 4/2 + 4/4 + 4/2 + 8/2,

gives us nine terms. Then, combining like terms together, we get

(2/8 + 2/4 + 2/4) + (4/4 + 2/2 + 4/4) + (4/2 + 4/2 + 8/2) =
[(6/16) + (10/10) + (16/6)] = 32/32 nm,

but the result is a scalar number, not a vector; that is, we are multiplying points here (actually pseudopoints, because RNs have volume). This means, in the language of GA, that the multiplication operation doesn't affect the grade of the pseudoscalar, or 3-blade, anymore than it affects the grade of the scalar, or 0-blade.

So, while a vector times a vector is a bivector, a higher grade blade, and a vector times a bivector is a trivector, a higher grade blade, a scalar times a scalar is a blade of the same grade. Stating the same thing symbollically,

A¹ * B¹ = C² (in the form of the "sum" of the inner and outer product, the geometric product), and

A¹ * B² = C³,

for the vectors, and

A⁰ * B⁰ = C⁰ ,

for the scalars, but what's suprising is that

C³ * C³ = C³, not C⁹

for the pseudoscalars! This is because the three dimensions of the pseudoscalar are the internal dimensions of the volume, represented by the three terms of the RN, and the nine terms of the product are always just scalar expansions of their three terms, as I have shown above.

As far as I know, this has never been noted before, because GA isn't designed to use the 3-blades, except as what is called the unit trivector, symbolized by the upper case letter, I. This view of the trivector is a vector view, but it is also called a pseudoscalar, because its outer product commutes with everything, as a scalar does. Yet, unlike a scalar, it also changes sign under inversion! What does that remind you of?

What we have discovered here is that the dilation of the scalar, or origin of a coordinate system, in the form of the pseudoscalar, can be expressed in the usual terms of the three dimensions of vectors, as a trivector, or alternatively, in terms of the three "dimensions" of the RNs, as a pseudoscalar. Moreover, this leads us to view the binomial expansion, which is the dimensional expansion of duality, as the expansion of points, which don't expand; the expansion of vectors which expand vectorially, and the expansion of pseudoscalars, which expand scalarly. That is why the numbers down the left and right edge of Pascal's triangle are always 1, but the numbers between these two 1s are more than 1, but always symmetric.

However, there is a difference between the 1s down the left side, which are scalars (2⁰ = 1), because they have no direction, but magnitude only, and the 1s down the right side, which are pseudoscalars (2ⁿ), because they have all directions possible in n dimensions. Hence, the number of directions in the pseudoscalar is the same as the number in the scalar at n = 0, namely zero, but at n = 1, the number of pseudoscalar directions is two, represented by the scalar expansion in all the directions of a line from the perspective of the scalar point in the middle (linear dilation). At n = 2, the number of pseudoscalar directions is four, represented by the scalar expansion in all the directions of a plane from the perspective of the scalar point in the middle (quadratic dilation), and at n= 3, the number of pseudoscalar directions is eight, represented by the scalar expansion in all the directions of a cube from the perspective of the scalar point in the middle (bi-quadratic dilation), as clearly dipicted in Larson's Cube.

There is much more to say about all this, but suffice it to say for now, that these are concrete mathematical results that are the subject of the mathematical research at the LRC. The application of these scalar math concepts, in developing a suitable language for the LRC's scalar physics research, is a major goal of the LRC. The prospect of using the new language effectively is promising, given that string theory, the standard model's guage theories, and the fundamentals of group theory and symmetry principles, all converge to the same place: the space of the octonions.

It has long been suspected that the octonions have a deep connection with physics for many reasons, but so far the connection has escaped physicists. Now that there is a new kid on the block, though, and things may change relatively quickly. Our first objective is to understand the RSM, and we've made a lot of progress since first discovering it. Yet, none of this means anything, unless we can connect it to observations. We have hints on how to do that with photons of radiation, but our primary goal is to use the RNs to find the properties of neutrinos, electrons, positrons, protons, and neutrons, and how they relate to each other and photons, in atomic spectra.

If we can do that, we will be on our way, because the origin of the material properties of mass, charge, and spin, are unexplainable in vector physics, but they should be relatively easy to explain in scalar physics, where the three "dimensions" of RNs, correspond to outward motion in space, outward motion in time, and inward motion in space, or in time, the magnitudes of which are functions of events and their associated probabilities.

Thus, since forces, such as gravitational, electrical, and magnetic forces, are properties of motion, we expect to find discover how they are related to the three dimensions of motion in the RNs. An exciting prospect, indeed.

I guess this concludes my long-winded initial answer to antoniseb's question. If there are followup questions, I would be happy to try to answer them.

Excal

Hmmm. I'm not sure how to interpret the silence. I know this is primarily an astrophysics/cosmology forum, but the science of astrophysics and cosmology rests on the foundation of particle physics. The whole idea of big bang, cosmic inflation, nucleosynthesis, etc, is developed on the basis of the standard model and general relativity theories, so these fundamental sorts of things are hopefully seen as vital here. Again, the trouble with physics right now is that these two theories must be treated separately, as if the quantum and the continuum were unrelated, though we know that they are not, and this conflict emerges most clearly in today's concepts of cosmology and astrophysics.

However, to make my point a little more explicit and to explain things a little further, I'll refer to John Baez's characterization of Geoffrey Dixion's work in Week 59 of his online mathematics tutorials:

Baez admits tha