The Larson Research Center

The third post in the BAUT forum follows.  These last three posts are a continuation of the RST thread in the “Against the Mainstream” forum of BAUT.

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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.