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.

 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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.

 

The second post in the BAUT forum follows.  This and the last post are a continuation of the RST thread in the “Against the Mainstream” forum of BAUT.

 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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.

 

I’ve been asked to discuss the new RSM mathematics in the huge discussion forum at the Bad Astronomy and Universe Today website , so I thought I would repost it here due to time constraints.  The first post follows.  I will post the second one later.  My ID over there is Excal.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 

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

 

In the American Heritage Dictionary, mathematics is simply defined as “the study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.” In the beginning, mathematics was used for counting, measuring, bookkeeping, etc, but, since the days of Newton, it has also been used for modeling, using a system of differential equations. In applications of theoretical physics, finding the solutions to these equations is a challenge, but has mostly centered on three equations, or their equivalents: the diffusion equation; the wave equation; and the sine-Gordon equation.

These three equations have been studied extensively, and they form the basis of the LST community’s physical theories, including Maxwell’s equations for electromagnetism, Einstein’s equations for gravity, and Schrodinger equation for quantum mechanics. Not suprisingly, since the equations of these systems involve continuous quantities, their adaptation to the development of theories of quantum physics has worked only to a limited degree, reflected in the fundamental crisis of theoretical physics that we are currently discussing in the The Trouble with Physics blog on this site.

Of course, if energy and matter are quantized, as experiments show they are, this suggests that the use of a discrete system of some kind, as opposed to a continuous system of differential equations, might be a more suitable approach to use, in seeking a solution to the fundamental crisis of physics. In fact, Stephen Wolfram, in his recent book, New Kind of Science, declares that his results show that a discrete approach would greatly simplify the task. He writes:

One of the most obvious differences between my approach to science based on simple programs and the traditional approach based on mathematical equations is that programs tend to involve discrete elements while equations tend to involve continuous quantities.

But how significant is this difference in the end?

One might have thought that perhaps the basic phenomenon of complexity that I have identified could only occur in discrete systems. But from the results of the last few sections, we know that this is not the case.

What is true, however, is that the phenomenon was immensely easier to discover in discrete systems than it would have been in continuous ones. Probably complexity is not in any fundamental sense rarer in continuous systems than in discrete ones. But the point is that discrete systems can typically be investigated in a much more direct way than continuous ones.

Wolfram’s thesis is that it’s not only easier to study nature via a discrete system, but that it’s possible to conceive of an entirely new system of physical theory, which is not based on the traditional system of mathematics, but on a new system of mathematical rules. He writes:

If theoretical science is to be possible at all, then at some level the systems it studies must follow definite rules. Yet in the past throughout the exact sciences it has usually been assumed that these rules must be ones based on traditional mathematics. But the crucial realization that led me to develop the new kind of science in this book is that there is in fact no reason to think that systems like those we see in nature should follow only such traditional mathematical rules.

What has led Wolfram to this conclusion is his discovery that very simple computational algorithms can lead to “behavior that [is] as complex as anything I [have] ever seen.” Though many might have missed the fundamental significance of this discovery, it’s impact on Wolfram was profound:

It took me more than a decade to come to terms with this result, and to realize just how fundamental and far-reaching its consequences are. In retrospect there is no reason the result could not have been found centuries ago, but increasingly I have come to view it as one of the more important single discoveries in the whole history of theoretical science. For in addition to opening up vast new domains of exploration, it implies a radical rethinking of how processes in nature and elsewhere work.

Coming from Wolfram, whose academic and scientific accomplishments are legendary, and who has made a fortune by exploiting the principles of traditional mathematics, and his ability to write software to formulate and solve complex equations from symbolic input, this is not a case of idle speculation. Like Larson, he understands that new ways of thinking are what are needed to successfully advance the scientific knowledge of mankind beyond today’s science. He confesses:

It is not uncommon in the history of science that new ways of thinking are what finally allow longstanding issues to be addressed. But I have been amazed at just how many issues central to the foundations of the existing sciences I have been able to address by using the idea of thinking in terms of simple programs…Indeed, I even have increasing evidence that thinking in terms of simple programs will make it possible to construct a single truly fundamental theory of physics, from which space, time, quantum mechanics and all the other known features of our universe will emerge.

However, as might be expected, the LST community is not very receptive to this new approach. Nobel prize winner Steven Weinberg, in reviewing Wolfram’s book, in an article entitled, “Is the Universe a Computer?” writes:

Wolfram himself is a lapsed elementary particle physicist, and I suppose he can’t resist trying to apply his experience with digital computer programs to the laws of nature. This has led him to the view (also considered in a 1981 article by Richard Feynman) that nature is discrete rather than continuous. He suggests that space consists of a network of isolated points, like cells in a cellular automaton, and that even time flows in discrete steps. Following an idea of Edward Fredkin, he concludes that the universe itself would then be an automaton, like a giant computer. It’s possible, but I can’t see any motivation for these speculations, except that this is the sort of system that Wolfram and others have become used to in their work on computers. So might a carpenter, looking at the moon, suppose that it is made of wood.

While Weinberg sees no reason to suppose that Wolfram’s ideas represent a promising alternative to the mathematical equations of traditional science, where the structure of the physical universe is described in terms of continuous systems, this opposition is primarily justified by an attempt to show that Wolfram has jumped to conclusions over “just how fundamental and far-reaching” the consequences of his simple programs are. However, it’s quite evident that Weinberg is simply protecting his own turf as a particle physicist. He writes:

Wolfram claims to offer a revolution in the nature of science, again and again distancing his work from what he calls traditional science, with remarks like “If traditional science was our only guide, then at this point we would probably be quite stuck.” He stakes his claim in the first few lines of the book: “Three centuries ago science was transformed by the dramatic new idea that rules based on mathematical equations could be used to describe the natural world. My purpose in this book is to initiate another such transformation….

Nevertheless, Weinberg uses the very same word “stuck” to characterize the state of theoretical physics in his own book, Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature, so the issue is not a matter of the efficacy of traditional science. Everyone agrees that modern physicists are “stuck” and that they are finding it very difficult to advance the laws of physics towards a “final theory,” in which the concepts of discrete and continuous phenomena are unified into one physical theory. Hence, while Weinberg may be very skeptical of Wolfram’s approach, the motivations for seeking a discrete system solution to the theoretical crisis should nevertheless be obvious to everyone.  Certainly, the obstacles to considering discrete systems in physics cannot be based on a lack of merit in the notion of discrete systems themselves, since the structure of nature is undeniably discrete in a very real sense.

On the other hand, in recognizing the discrete nature of physical phenomena, there is nothing to restrict us to considering only the rules of discrete forms of simple programs, like cellular automata. It’s important to recognize that natural numbers themselves are discrete and the algebra of their mathematical operations exhibits the same properties as do the interactions of physical objects; that is, when various numbers are combined into greater composite numbers, or existing combinations of numbers are separated into their lesser constituent numbers, the new combinations, or constituents, are totally predictable, given the parameters of the operations acting upon them.

While the vectorial motion of objects is clearly continuous, so that a differential equation gives a relation between the value of some varying quantity and the rate at which that quantity is changing, and perhaps the rate at which that rate is changing, and so on, this means that it is the relationships of the continuously changing numbers represented by the equation that are important, and it is the cause of that change, a force, or a quantity of force, an acceleration, that some of these numbers represent, that is ultimately what we seek to understand in terms of identifying the laws of nature.

For example, if we have a solution to a system of differential equations that can predict the path and energy of a particle through a known environment, given its velocity, mass, charge, spin, etc, using the correct numbers, this means that we can say something definitive about how these properties relate to one another in determining the universal behavior of physical phenomena, and we can classify them accordingly. Indeed, this has been the grand goal of the traditional program of physics, inaugurated by Newton himself.  The description of the structure of the physical universe, in terms of a few fundamental interactions among a few elementary particles, culminates in the finest intellectual achievement of Newton’s program of research, called the standard model of particle physics.

However, when it comes to the physics of discrete entities in the microcosmic realm of the quantum world, ordinary numbers are not sufficient. The solutions to the equations not only have negative roots, which has given rise to the idea of antimatter, but it is impossible to describe the physical phenomena involved, using these equations, without employing complex numbers, which are indispensable, given the concept of uncertainty; that is, we not only need differential equations to deal with discrete phenomena on a continuous basis, but we also need to use imaginary numbers in these equations.

This is a significant point, because the idea of imaginary numbers is an ad hoc invention of the human mind, originally devised to deal with negative numbers (as the square root of -1). In fact, without imaginary numbers, the traditional idea of higher dimensional numbers, such as complex numbers (n¹), quaternions (n²), and octonions (n³), would not be possible (except with Hestenes’ GA, where ‘i’ has a geometric significance).  All numbers would be limited to the set of positive, real numbers (n⁰), and, needless to say, this not only would make the LST community’s formulation of quantum mechanics impossible, it would make all of LST physics impossible, because it would make the concept of vectors impossible.

This is not hard to understand, because it’s the imaginary number that gives numbers direction, that turns these discrete quantities into a something that can be used to define more than simple quantity. With the imaginary number ‘i’, a number can be written that represents a point in any direction from 0; that is, a complex number is not just a point on a number line, that is greater than, or less than, some other number on the line. However, modern mathematicians are quite amazed that the human invention of the imaginary number by mathematicians would turn out to be indispensable to the quantum mechanics of physicists, centuries later.

Yet it’s not improbable that school children of future generations will ask why the mathematicians of our day were so amazed, because to them it will be very clear that direction is a property of motion, and, since physicists study motion using numbers, numbers clearly should have a direction property too.  However, the insight the children will have then, which modern mathematicians and physicists don’t have today, is that numbers do have direction. In fact, they have always had the direction property that was needed, and it had been used routinely in other contexts since the dawn of civilization; there was really no need to invent the “imaginary” number to “give” numbers the direction property.

The truly amazing thing to contemplate is what we claim here at the LRC: Recognizing the inherent property of direction that numbers possess, without invoking the magic of imaginary numbers, revolutionizes the mathematics of physics, transforming it from a continuous system into a discrete system from which the continuous system, and its real numbers, emerge naturally.