## Sunday, March 12, 2017

### BASIC CONCEPTS OF THE CALCULUS OF DIMENSIONAL DISTINCTIONS Installments 1 through 4

INTRODUCTION TO THE MATHEMATICAL CONCEPTS UNDERLYING THE TRIADIC DIMENSIONAL VORTICAL PARADIGM (TDVP)

A HISTORICAL NOTE:
This is an update of material that was originally written in 2011 and 2012, as part of “Space, Time and Consciousness”, a manuscript intended for publication in 2013 or 2014. Due to a series of tragic events that seriously affected my life and my ability to work, publication of this material was indefinitely delayed. We still intend to publish this material when we can, but since a number of people following my Transcendental Physics blog have asked for an introduction to the Calculus of Dimensional Distinctions (CoDD), I am posting the basics on this blog. – Edward R. Close, March, 2017

Inserted March 18, 2017:

.
HOW COULD THE ANCIENTS HAVE MOVED HUGE BLOCKS OF STONE THAT WOULD BE VERY DIFFICULT IF NOT IMPOSSIBLE EVEN WITH MODERN TECHNOLOGY?

These installments contain the story of how gimmel, the third form of reality was discovered, and how it is necessary for any stable life-supporting physical structure to exist. They also explain why gimmel is not directly measurable as mass or energy, and that the three forms of reality, mass, energy and gimmel are interchangeable under certain circumstances.  The stuff of physical reality consists of matter (mass) energy and gimmel. All three exist in every atom. We know how to convert mass to energy, and energy to mass. If we learn how to convert mass to gimmel, and can convert a large part of the mass of an object to gimmel, which has no mass, heavy objects can be made to become very light temporarily. If someone in the distant past could do this, huge blocks of stone, with most of their mass temporarily converted to gimmel could be moved easily and reconverted once in place.

NOTE: Some additional changes and clarifications have been added elsewhere in the first four installations

INSTALLMENT  #1

THE LAWS OF FORM AND THE CALCULUS OF DIMENSIONAL DISTINCTIONS

"A universe comes into being when a space is severed…" G. Spencer Brown, “Laws of Form” A Note on the Mathematical Approach, Page v, The Julian Press, New York, 1972

One Sunday in 1979, the author and his wife Jacqui were enjoying lunch at the Inn of the Seventh Ray in Old Topanga Canyon, near Los Angeles, California. After lunch we strolled out to the little bookstore behind the restaurant.  Jacqui picked a small book from a shelf, handed it to me and said: “I think you need to read this book". The book was G. Spencer Brown's “Laws of Form”. I was immediately taken with the clarity and elegance of Brown's presentation of the logic of the calculus of indications. His calculus revealed an entire world of logical representation prior to, and thus forming the actual basis of, all mathematical representation and symbolic logic. At the time I had only completed my degree in mathematics with a physics minor, one year of a MS degree program in theoretical physics and was doing independent research. Finding Brown’s work at that time was serendipitous. Laws of Form, especially the calculus of indications, would play a major role in the development of the ideas that led to a paradigm shift to a consciousness-based reality.

About ten years later, while working on my second book, “Infinite Continuity”, I discovered that Brown's calculus of indications (CoI) could be adapted to describe our perceptions and conceptions of matter and energy interacting in space, time and consciousness in a way that would allow us to expand the current paradigm to include consciousness in the equations. I saw reflected in the theorems of the Laws of Form the underlying logical patterns of reality. The calculus of distinctions (CoD) was introduced in Infinite Continuity and used to test a number of physical and cosmological hypotheses. It was used again in my third book, “Transcendental Physics” to prove the necessity of the existence of a non-quantum receptor prior to the emergence of the first physical wave and/or particle out of the big bang.

After Infinite Continuity was published in 1989, and again after Transcendental Physics was published in 1997, I was asked: why is it called a calculus? And, how is the CoD different from the CoI in the Laws of Form? I will answer those questions here. First, a calculus is any system of symbolic representation that allows transformation of meaningful statements of mathematical logic from one form to another, using operations based on one or more axiomatic relationships, called the primary equations of the calculus. The further purpose of this discussion is to clarify the definitions, meaning and applications of the extension of the CoD, the Calculus of Dimensional Distinctions (CoDD) used in the TDVP paradigm shift. In the process, the differences between this calculus and Brown's CoI will be explained.

Both the CoI and the CoDD use a simple symbol to represent or indicate the perception or conception of a distinction. In Brown’s CoI, the symbol that was used to indicate a distinction is . I have used the symbol n in the CoDD, where the subscript n indicates the dimensionality of the distinction. I chose a curved symbol because most distinction boundaries are curvilinear in form, not rectangular. Conceptually, this difference is appropriate because we deal primarily with variations of curved ovoid, or vortical forms in TDVP. The CoDD is different than the CoI both conceptually and operationally, in a number of very significant ways.  Important differences arise principally from the way the concepts of consciousness, existence and dimensionality are treated. Let’s look at each of these concepts as they are represented in the CoDD in contrast to the way they are treated in the CoI.

The existence of a conscious observer is given as a priori and separate from objective reality in Brown’s CoI, in much the same way it is in conventional mathematics and physics. The existence of something or someone, independent of physical reality, existing only in the role of observer, capable of drawing conceptual distinctions is assumed. This is a natural assumption since the experience of every self-aware sentient being, conscious of three dimensions of space and one point in time, is of the existence of self and the apparent existence of “other”, through the physical senses, creating mental images of a world outside of self. Furthermore, the apparent lack of control and randomness in the dynamics of the outer world, suggests that the other is completely independent of the observer. This, however, is demonstrably not true.

There are now at least three types of evidence strongly suggesting that this sense of complete independence is an illusion: 1.) The six-sigma probability evidence of meta-analyses delineated in Reality Begins with Consciousness (RBC) that establishes the existence of psi phenomena such as remote viewing, precognition and psycho-kinesis. 2.) Quantum mechanical evidence of the direct interaction of the observer with the physical character of elementary wave/particle phenomena, delayed-choice and relativistic time phenomena, quantum entanglement, and non-locality. 3.) The hidden connection of phenomena that appear unrelated in the 3S-1t world detected by our physical senses through additional dimensions. Prime examples are the propagation of light and the explanation of gravity in the general theory of relativity.

So, in CoI the existence of self is given, but the existence or non-existence of any distinction perceived or conceived, is not considered to be an important issue. (Ref. Laws of Form, page 101). Whether a given distinction represented symbolically actually exists as a real object or not is unimportant in the application and logic of the CoI, while, in the application of the CoDD in the Triadic Dimension-Distinction Vortical Paradigm (TDVP), the concept of existence is extremely important. Given that there is a reality that we perceive through the physical senses, and possibly through other faculties of consciousness that are poorly understood in the current scientific paradigm, the existence of stable and persistent forms and the distinctions that make up those forms depend upon the relative electrodynamics and dimensionality of the form and the observer. In other words, the perceived form of an object is relative. This prompted us to coin the term “indivension” in Reality Begins with Consciousness (RBC). If perception is relative, and complete independence is an illusion, is there never an objective reality completely independent of individual consciousness? In the N-dimensional reality posited by TDVP, complete reality would only be fully perceived through an awareness of all finite dimensions from a transfinite or transcendental consciousness. If reality exists in more dimensions than we are capable of perceiving, we are not aware of the totality of reality, only the parts or aspects of it that we are capable of experiencing through the drawing of distinctions in three dimensions of space and one dimension of time. Definition and exploration of additional dimensions and the relation of consciousness to those extra dimensions is the central thrust of this discussion.

The logic of the CoI applies to any distinction equally well, regardless of the dimensionality or type of the distinction. Because of this, dimensionality is not treated as such in the Laws of Form. Brown does mention dimensionality in relation to modes of expression: on page 92 in ‘Notes on Chapter 6’ he says: “we may observe that, in [CoI] expressions, the mathematical language has become entirely visual, there is no proper spoken form, so that In reverbalizing it we must encode it in a form suitable for ordinary speech. Thus although the mathematical form of an expression is clear, the reverbalized form is obscure.

“The main difficulty in translating from written to the verbal form comes from the fact that in mathematical writing we are free to mark the two dimensions of the plane whereas in speech we can mark only the one dimension of time.

“… in ordinary speech, to avoid direct reference to a plurality of dimensions, we have to fix the scope of constants such as ‘and’ and ‘or’, and this we can do most conveniently at the level of the first plural number. But to carry the fixation over into the written form is to fail to realize the freedom offered by an added dimension.”

Relativity theory has established time as the fourth dimension; but, even though time is discussed in Chapter 11 (page 62) of the Laws of Form, it is not linked specifically to dimensionality. The introduction of time and the CoI equivalent of imaginary values appear as “departures from the form” in Laws of Form (p. 58 – 68). Representation of time becomes very complex and circuitous in the CoI, because the CoI applies only to basic distinctions of human perception which occur in one, two or three dimensions, the inclusion of time is conceived as a “departure “from the form indicated by the initial equations of the calculus. In the CoI (pages 99 -100, Laws of Form) Brown discusses the departure as “subversion” (self-referentiality) of CoI equations of degree 2 and higher. He says:

“Any evenly subverted equation of the second-degree might be called alternatively evenly informed. Such an expression is thus informed in the sense of having its own form within it and at the same time informed in the sense of remembering what has happened to it in the past.

“We need not suppose that this is exactly how memory happens in an animal, but there are certainly memories, so-called, constructed this way in electronic computers…

“We may perhaps look upon such memory in this simplified in-formation as a precursor of the more complicated and varied form of memory and information in man and the higher animals. We can also regard other manifestations of the classical forms of physical or biological science in the same spirit.”

He goes on to discuss how second degree equations developed in Laws of Form mimic real world physical forms and processes:

“Thus we do not imagine that the wave train emitted by an excited finite echelon [a CoI equation of the second degree] to be exactly like the wave train emitted from an excited physical particle. For one thing the wave form from an echelon is square, and for another, it is emitted without energy. We should need, I guess, to make at least one more departure from the form before arriving at a conception of energy on these lines. What we see in the forms of expression at this stage although recognizable, might be considered as simplified precursors of what we take, in physical science, to be the real thing. Even so, their accuracy and coverage is striking.”

In the CoDD we expand the concept and notation of the calculus to include symbolic indication of the number of dimensions associated with the distinction, i.e. the dimensional domain within which the distinction is drawn. This allows us to move more easily into applications of equations of higher degree, makes the calculus more powerful for the analysis of logical statements, and allows more accurate mathematical representation of physical structure and processes, and functions of consciousness like cognition and memory.

Like most scientists and mathematicians, Brown expresses some surprise that logical structures revealed by the laws of form appear to mimic the patterns and observations of what we take to be reality in the physical, biological and psychological sciences. (Ref. page xxii, Laws of Form) By contrast, in the TDVP there is no basis for surprise that the calculus, like any other valid form of mathematics and symbolic logic, reflects the structure of reality, since the basic logical structure of distinct forms in reality is the actual source of the form and structure of the calculus, and we can formalize this concept as a basic axiom. This allows us to include consciousness and the functions of consciousness in the equations of science. Something that, in the history of modern science, to my knowledge, has never been done before.

The CoDD, like Brown’s calculus, is a system of mathematical representation logically prior to conventional forms of symbolic logic and mathematics. Its scope and operational power extends beyond that of the CoI, which is considerable, because of the explicit inclusion of dimensionality and existential consciousness.

2ND INSTALLMENT of Intro to CoDD:

The Axiom of Logical Consistency:

All of Reality, including Time, Space and Consciousness, matter, energy and all Aspects of the Physical Universe, conform to a Consistent Universal Mathematical logic.

This is, after all, the basis of all natural science. Albert Einstein’s famous statement “I want to know God’s thoughts, the rest is just details.” reflects this point of view. Whether you think of the logical structure of the universe as “God’s thoughts” or the natural unfolding of reality, a logical consistency is assumed.

Note that this axiom applies to the various branches of mathematics, including geometry and number theory, and to the various fields of science, and in this way, unites them all as one consistent logical system.

It is also important to note that the logical system of reality is not a closed system. In keeping with Gӧdel’s Incompleteness Theorem, questions that cannot be answered within the system as we know it, are answerable in an expanded version of the system. In this way, reality is an infinitely expanding logical system. The acceptance of reality as an unbounded logical system allows inclusion of consciousness as a real part of reality to which the CoDD can be applied.

The lack of interdisciplinary consistency found in conventional science and mathematics is a direct result of the arbitrary academic separation of the branches of scientific endeavor, and the arbitrary separation of the consciousness of the observer from the object of observation. The CoDD, on the other hand, unites the various fields of science in a logical mathematical system, and involves consciousness from the very beginning and throughout the process. Quantum physics tells us that the “observer” is actually involved in bringing the form of reality into manifestation through the drawing of distinctions, and a more appropriate term, “participant” is suggested. By relating consciousness to reality, and confronting the deep question of existence versus non-existence, we may expand the initial equations of the CoI to accommodate dimensionality.

The importance of dimensionality was recognized by Hilbert and Minkowski, who introduced time as a fourth dimension, mathematically analogous to spatial dimensions. Einstein resisted this as “unnecessary mathematical sophistication” at first, but eventually accepted it as useful, and even necessary in the General Theory of Relativity. Einstein and other physicists proceeded to explain the action-at-a-distance force of gravity as a warping of the space-time continuum by matter and energy. In TDVP, we carry this reasoning forward, applying it to the association of additional action-at-a-distance forces with additional dimensions. There is, however, a very subtle and very important point to be made:

In 1952, Einstein added a fifth appendix to his classic book, “Relativity, the Special and General Theory, a clear explanation that anyone can understand”. Appendix V was titled: “Relativity and the Problem of Space”.  In a note to this appendix, he says:

“I wish to show that space-time is not necessarily something to which one can ascribe a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept of empty space loses its meaning.”

What?!! One might ask: How can we explain a fundamental force of nature, a physical force, acting over distance, without the benefit of a medium of transmission, as the warping of space-time, if space-time is not an objective “something” independent of matter and energy? How can something without an independent existence be warped or bent? The key to understanding this is in Einstein’s words “objects are spatially extended”. One must avoid the temptation to think of space-time as a kind of ether or medium through which electromagnetic waves and gravitational forces are transmitted. Instead, space-time is an integral part of the extended multi-dimensional fabric of reality. Space-time is not something existing apart from matter, energy and consciousness; it exists because of matter, energy and consciousness. This is why the Michelson- Morley experiment failed to reveal the existence of ‘ether’ a medium for the propagation of light.

Einstein enabled us to realize that the reality experienced by any given conscious being is relative, depending upon the dimensional reference frame, mass and motion. From quantum physics, we now know that the observer is a participant. The experience of reality is also affected by the limitations of the sensing apparatus. The drawing of distinctions, involving conscious choices, can affect the manifestation of reality at the quantum level. From the beginning, we must recognize this inter-dependence of consciousness, matter/energy and dimensionality. In the CoDD, as we develop the process we call ‘dimensional extrapolation’, we will see how dimensionality also affects the way reality is perceived and experienced.

Science arises from the desire to understand the nature of the reality we experience and the need to cope with life causes us to develop a pragmatic reductionist approach to understanding reality. We must focus on the problem at hand, whether it is, e.g., how to feed our children or how to avoid being injured or killed by a tiger or a bus. Similarly, science has attacked the problem of understanding reality by trying to separate various aspects and components of reality for study. The different fields of science arose from efforts to understand specific aspects of reality, physical, chemical, biological, psychological, etc. A comprehensive paradigm, however, must reverse this tendency toward reductionism and find a way to integrate all aspects of reality into one consistent paradigm. Recognizing the underlying role of dimensionality in perception provides a logically and mathematically consistent way of doing this without negating the knowledge and understanding obtained through the reductionist method.

The strength of the TDVP mathematical logic approach lies in its capability of describing all tangible aspects of reality. By basing the CoDD on the most basic triadic processes of perception, the logical patterns of reality experienced by sentient beings are reflected in the resulting mathematical structure. The shortcoming of conventional mathematics is that, prior to G. Spencer Brown’s Laws of Form, accounts of mathematical logic and its applications always started in the middle of the story. Brown starts with primitive concepts that underlie all of the basic mathematical concepts and operations. The CoDD expands the logic of the CoI logically and operationally, and allows us to apply the same rules of logic to an n-dimensional reality. Thus the CoDD spans number theory, geometry, physics and consciousness by combining mathematical, geometric, physical, and conceptual representation in one symbolic formalization with appropriate operational rules.

Our current mathematics education usually starts with an introduction to the fundamental operations of arithmetic: addition, subtraction, multiplication and division. Little time if any, is spent on the underlying concepts of continuity, discreteness, infinity, enumeration and equivalence. Still underlying these concepts are the even more basic concepts of distinction and indication. These deeper basic concepts are considered to be too abstract and unnecessary for the development of useful math skills. It is generally considered more practical and efficient to start with the four “fundamental" operations. The overlooked more basic concepts however, are closely related to the way the human mind works and the nature of the underlying reality reflected in it. This correspondence of infinite continuity, distinction, indication, discreteness, enumeration, and equivalence to reality and the basic functioning of human thought is the reason students are able to comprehend addition, subtraction, multiplication, and division in the first place, and carry on to the more complex mathematics of analytical geometry, algebra and calculus without recognizing the role of the more subtle underlying concepts. Similarly, a formal education in the physical sciences generally by-passes the underlying concepts arising from mind-matter interaction and borrows mathematical tools developed by number theorists, to analyze data arising from physical experiments.

With the CoDD, we go back to the most primitive basic concepts, so that mathematics is re-connected with its space-time-consciousness roots, and then, through application of the CoDD to quantum and relativistic physics at the elementary particle, or quantum level, reconnect physics with its matter-energy-consciousness roots. In this way, the CoDD re-unites mathematics and physics and relates them to consciousness as the primary drawer of distinctions, the collector and processor of data, and the primary organizer of data into meaningful information. This re-connection allows us to include the functioning of consciousness from the very beginning so that it is not an excluded concept when we reach the level of understanding necessary to produce a comprehensive paradigm. In this presentation, I will try to use conventional mathematical and physics terminology as much as possible, and relate new or slightly different concepts to conventional thinking by analogy.

The first step, before we can develop the CoDD to the level of sophistication that we can use it to describe the reality we experience, we must define our terms, concepts and processes in a logically rigorous manner. There are three levels of distinctions: perceptual, conceptual and intentional. Within each level, there are three types: extent, content and intent. Within each type are three forms: linear, areal and volumetric. The triadic nature of reality which begins with the first distinction: the conscious entity, the object distinguished, and the rest of the world that it is distinguished from, appears to be propagated throughout reality as the primary structural feature.

DISTINCTIONS OF EXTENT
Let indicate the drawing of a distinction, and let A  B describe the situation in which A indicates something that is distinguished from B. For example, A might be the area within a rectangle or a circle and B the rest of the two-dimensional plane upon which the figure is drawn. Or A might be the volume of a sphere and B the rest of space. In general A is the content of the distinction and B is the rest of the universe.

Now let us analyze this symbolic representation of distinction thoroughly: ‘A’ represents that which is distinguished; ‘ ‘represents the edge or boundary of the distinction, and ‘B’ represents that from which it is distinguished. No distinction is completely described without this triad. But this is not the whole story. The story is not complete without addressing the relationship between consciousness and the drawing of distinctions. Accepting the abundant demonstration of the truth of the Copenhagen interpretation of quantum mechanics, neither this triad of symbols, nor the reality they represent have any real existence or meaning without a conscious receptor. At the quantum level, reality remains in the probabilistic state of multiple possibilities until this triad is completed by a sentient being in the drawing of a distinction. Thus we have another significant triad: 1. Reality, 2. sentient being, 3. symbolic representation or map of reality. The symbolic representation in the CoDD is connected to the reality it attempts to describe or map by the consciousness of the sentient being creating the map. This process is analogous to the original action of Primary Consciousness, drawing the first distinctions of matter and energy at what we have called the “event horizon” of the big bang creation of the physical universe in our previous book: “Reality Begins with Consciousness”.

The process of the drawing of the distinction by a sentient being proceeds as follows:

(1.)  Perception, (2.) conception (3.) Representation.

The first distinction drawn by a conscious being, the distinction prior to, and necessary for all subsequent distinctions, is the distinction of self from other. Once this distinction is drawn, subsequent distinctions may then be drawn in both domains: self and other. These distinctions constitute the “reality” known by each individual sentient being.

I plan to add more to this post weekly, or as time permits, to provide a reader with at least a basic understanding of the Calculus of Dimensional Distinctions and its applications.

3rd INSTALLMENT:

Definition: The term ‘Domain’ means a multi-dimensional domain within which distinctions can be drawn.

This term combines the sense in which it is used in number theory to include the concepts of a  ring, set or field of numbers, and the sense in which it is used as a geometric or spatial concept. The key term in this definition is the word “dimension”.

Definition: A dimension is a distinction describable and measureable in variables of extent.

Space, time and consciousness are dimensional domains. Matter, energy and thought are not. They are distinctions of content. We have also introduced the term dimensionometry to extend the concepts of geometry beyond three dimensions.

The drawing of distinctions is literally a double-edged sword: it allows us to operate as individual sentient beings, but it also obscures the deeper truth of the underlying unity of reality. The evidence of non-locality in quantum physics suggests that reality is infinitely and continuously connected, but, because our physical senses are limited to a specific scale and range of perceptions in three dimensions of space and one point in time, within which we draw finite distinctions, reality appears to be divided up into more or less independent parts. The underlying unity is revealed, however, when we find that all of the distinctions that make up our individual realities are drawn in the same infinitely continuous space-time-consciousness continuum. The logical patterns and forms revealed by the CoDD are the patterns and forms projected from the innate logical structure of the infinite continuity of the Primary Receptor, necessarily present at the time of any proposed origin event, and reflected in the finite conceptual realities of individualized consciousness.

So, what exactly is the CoDD, and how does the calculus reveal the innate structure of reality? The calculus reveals the connection between the structure of the space-time continuum and the structure of consciousness by mimicking the patterns created by their interaction. This connection will become clear through dimensional extrapolation; i.e. the conceptual movement from one dimensional domain to another.

The CoDD consists of a set of rules and procedures governing the formal and logical mathematical manipulation of symbols representing conceptual distinctions drawn from perceptions of reality. The process of the drawing of distinctions is the real basis of all perception and conceptualization leading to cognition and understanding. Thus developing a CoDD is tantamount to defining the logical structure underlying all cogent thought, including all branches of science and mathematics.

In the Laws of Form, the calculus of indications is based on two initial equations. An equation is a basic statement of equivalence. The logical structure of an equation is analogous to the logical structure of a sentence in any language. If X = A + B, e.g., the left side of the equation is analogous to the subject, the equals sign is the verb, and the right side is the predicate or object. An equation can be reflexive, in which case it can be read either way, left to right or right to left. In such a case, the equation is said to be an identity. If the two sides are identical, the equals sign can be replaced by the symbol ≡ to draw attention to the essential identity of the two expressions.

The CoI initial equations are expressions of the drawing of the distinction of inside from outside, and the CoI symbol, , indicates crossing a boundary into the “inside” and indicates re- crossing back to the “outside”, undistinguished reality and the initial equations of G. Spencer Brown’s CoI were:

(1.)         ┐┐= , and   (2.)        =    .

These equations are based on the primitive instincts of consciously distinguishing inside from outside (analogous to distinguishing self from other) and the movement or changing of focus from inside to outside, or outside to inside, by the crossing of the boundary between them.

Note that the blank space on the right side of equation (2.) does not indicate zero or “nothing”; it simply denotes the lack of distinction.

Equation (1.) is reflexive. Every instance of ┐┐in a string of symbols can be replaced by and vice versa. Equation (2.), on the other hand, is not. While , which indicates the crossing and re-crossing of a distinction boundary, and thus can be replaced by a “blank”, i.e., removed from the string of symbols comprising an expression or equation, not every blank space indicates that a boundary has been crossed twice.

Equation (1.) is called the form of contraction, and Equation (2.) is called the form of cancellation. In calculation, a crossing symbol with a stroke through it (  ) can be written in place of the empty space, if the crossing and re-crossing of the boundary needs to be noted or remembered.

The symbols and equations in the CoI do not directly indicate dimensionality or the type (extent, content or intent) of distinction being drawn. Unlike the CoI initial equations, the CoDD initial equations include the concept and the indication of the dimensionality and type of each distinction, and accordingly are necessarily more numerous and somewhat more complex. These adaptations that distinguish the CoDD from the CoI were made in order to include quantization and dimensionality in the equations, concepts crucial to the expansion of mathematics to include the functioning of consciousness, and to the understanding of its relationship to the nature of  multi-dimensional reality, however many dimensions there may be.

Since we can draw distinctions of zero, one, two, three, four, and perhaps more dimensions, the dimensionality of a distinction can be indicated by a numerical subscript.

In general:
n specifies a distinction of n dimensions, where n is an integer ≥ 0. Now, if n represents a distinction of extent, as opposed to a distinction of content, then the symbol0 denotes a dimensional singularity or point.

Definition: Projection –The Projection of a geometric figure out of its n-dimensional domain creates an (n+1) -dimensional domain.

Thus the projection of a singularity produces a line, the projection of a line produces a plane, the projection of a plane produces a volume, the projection of a volume produces an event, and the projection of an event produces a timeline. The concepts of event and timeline will be further defined and explained when we apply the CoDD to n-dimensional domains with n ≥ 3.

This adaptation of the calculus to include dimensionality requires some modifications of the initial equations. The new CoDD primary equations are as follows:

3.)              mn  = nm =n, if and only if n = m ≥ 0
3.a)      If n > m, mn  = m + n  = mnn. (Note that m   is analogous to in the CoI notation.)
3.b)      If m > n, mn  = m + n  = mnm

4.)          mn   =     , or  if m = n

4.a)            mn    =  n, if n > m ≥ 0

Also, mn  is not possible if m > n, since an m-dimensional distinction cannot be contained in an n m -1 dimensional distinction. For example a three-dimensional sphere cannot be contained within a two-dimensional plane.

Notice that Equations (3.) and (4.) are commutative, but (3.a), (3.b) and (4.a) are not. This will have significance when we apply the CoDD to distinctions of content.

Dealing with the symbol for "no distinction" brings up an interesting point: Zero is sometimes confused with the concept of nothing, and "no distinction" could easily be confused with zero. It is important to note that the symbol ⦰ as used in the CoDD represents the absence of distinction, which is not the same as zero, or a state of nothingness. So we have three concepts that should not be confused:

Mathematically, zero is a point on the real number line between the positive and negative numbers, In the CoDD, the symbol ⦰ represents the cancellation of a specific distinction by the deliberate reversal of the conscious process that created it; a state of nothingness, on the other hand, is not a state open to experience or comparison because any comparison involves at least one distinction, which would destroy the state of nothingness.

The assumption that a state of nothingness can exist is not a scientific hypothesis because it cannot be falsified.

4th Installment

THE CALCULUS OF DIMENSIONAL DISTINCTIONS
Before we continue in this introduction to CoDD let’s refresh our understanding of what a calculus is. As stated in the first Installment, in general, a calculus is any system of symbolic representation that allows transformation of meaningful statements of mathematical logic from one form to another, using operations based on one or more axiomatic relationships, called the primary equations of the calculus. To clarify: calculation and transformation, as used in this statement, are one and the same thing. For example, the calculation 1+1= 2, is a logical operation (addition) that transforms the symbols 1+1 into a different, but equivalent symbol: 2. An algebraic operation is also a calculation, e.g., (x+y)2 = x2+2xy+y2, where the symbolic statement (x+y)2 is transformed into a  different equivalent symbolic statement: x2+2xy+y2. In the calculus developed by Leibniz and Newton, differentiation and integration are calculations that transform mathematical statements called functions that reach limiting values as an independent variable or variables approach zero.

DISTINCTIONS, DOMAINS AND EMBEDDED DOMAINS
In the CoDD, represents a distinction of extent and, in general, n represents an n-dimensional distinction. This means that 0 is a dimensional singularity or point, 1, a line, 2, a plane, and 3, a volume. Thus n represents an n-dimensional domain, capable of containing an infinite number of distinctions of extent. For example, a three-dimensional domain, 3, can contain an infinite number of 2 domains, 2 can contain an infinite number of 1 domains, and 1 can contain 0 an infinite number of times. This is an invariant characteristic of dimensionality that can be generalized as:
The Hypothesis of Embedded Domains: An n-dimensional domain, n, contains an infinite number of n-m dimensional domains for all dimensional domains with n ≥ m ≥ 1.

We can see that this is true for 0 ≤ n ≤ 3 by visualizing points within a line, lines within a plane, and planes within a volume, but it may not be immediately obvious that it is true for n > 3, because we have difficulty visualizing domains of dimensionality > 3. It will be a worthwhile exercise to prove this hypothesis using the CoDD, because it will serve as an example of the use of the CoDD to prove falsifiable hypotheses. In addition, the concept of embedded domains is a central feature of the concept of Dimensional Extrapolation, the movement from the reference frame of an n-dimensional domain to an (n+1)–dimensional domain, so we will take a brief side trip to prove it. But first we must interpret the CoDD as an operational tool for application to any statement that can be made within any logical system by defining basic logical statements in the language of the CoDD.

It is necessary to insert a note about notation here: There is no provision in Word for nested distinction symbols, and when they are created and imported into a Word or PDF document, they appear as a blank rectangle when the file is copied into this blog. (Jacqui tells me that I have to create all of the variety of nested symbols I need as separate j-peg files and import them as pictures. But that will take a lot of time, so I am representing nesting in the CoDD equations I need here by placing the nested symbol to the left of the symbol under which it is nested and reducing it to the size that would fit under the symbol. Thus mn indicates that m is nested in n, and ﬧﬧ indicates that and are nested in .

Note also that in the representation of logical statements and arguments, dimensionality subscripts may be dropped to simplify CoDD expressions only if all of the distinctions in the statement can be expressed in the same dimensional domain.

The Basic Statements of Logic in the CoDD:

The first distinction, without which no further distinction can be drawn, is the distinction of self from other. The initial equations of the CoDD, expressing the most basic conscious experience, the distinction of self from other, are:
mn        . (I)      and nn n    (II) where n m 0.

Equation (I) expresses the awareness of existing “in here” versus “out there” and Equation (II) expresses the awareness of equivalent experience. Readers familiar with G. Spencer Brown’s Laws of Form, will notice that I have reversed the order of the initial equations, relative to the analogous CoI initials. This is because Equation (I) represents the experience of self-awareness, which is necessarily prior to the awareness of equivalent experience, the basis of memory and logical sequential consistency.

Keep in mind that the blank on the right side of equation (I) can be replaced by the symbol ⦰ when we need to indicate or remember the state of non-distinction in the description of the an expression of a conscious experience of sequential events, or in a logical statement, and the state of non-distinction should not be confused with numerical zero or the concept of nothingness.  Also, the symbol in the initial equations of the CoDD indicates that they are reflexive, meaning that the transformation can be applied in either direction.

With these things in mind, we can adapt the CoDD for application to logic as follows:
Let A = n be a finite distinction of n dimensions, and assign the concepts “true” and “existential” to A. Then, if a CoDD statement, B, of dimensionality m, where m ≤ n, reduces to , through the application of the initial equations, B is also true, and if the statement B reduces to ⦰, implying that the opposite of is true, then B is the opposite of true, or false.

At this point, it may appear that with the true/false dichotomy, the CoDD is a simply a different mathematical/logical form of binary logic similar to Boolean algebra. But this is in fact not the case, because Boolean algebra applies to only three types of statements: True, False and Meaningless. The CoDD, like Brown’s CoI, must apply to four types of statements: the True, False and Meaningless of Boolean algebra, plus a fourth type, indeterminate, which is the logical equivalent of the imaginary numbers in arithmetic.

It should be clear that proof of Gӧdel’s Incompleteness Theorems requires that this fourth type must be included in any system of mathematical logic, because they show that within any finite system, logical statements can be made that cannot be proved true or false within the system. Such statements are not meaningless, they are simply indeterminate in the finite system within which they have been stated.

Many problems in pure mathematics, and in physics applications, notably electronics and computer science, cannot be solved without using imaginary numbers. G. Spencer Brown discussed this and its implications for symbolic logic in Appendix 2, pages 112- 135 in Laws of Form. The fact that he understood the importance of this is reflected in the first sentence of his preface to the first American edition of Laws of Form:

“Apart from the standard university logic problems, which the calculus published in this text renders so easy that we need not trouble ourselves further with them, perhaps the most significant thing, from the mathematical angle, that it enables us to do, is to use the complex values in the algebra of logic.”

In this quote, the term “complex values” refers to the numbers a + bi, the union of real and imaginary numbers (where i = Ö-1). In my opinion, the choice of the term “imaginary” to describe Ö-1, the square root of minus one, was unfortunate. The square root of minus one is no more imaginary than the real numbers. The CoDD shows us that appearances of imaginary numbers in calculations defined in an n-dimensional domain indicate that additional dimensions are involved, and that the problem is indeterminate in n dimensions. Proof of this is beyond the scope of the present discussion and must wait until we have defined the CoDD more completely.

Returning to the present discussion, we are translating the basic statements of conventional logic into the language of the CoDD. First, we can translate the logical statement “not A” as A . Proof: If A is true, A = , and by application of Equation (I), A = ⦰, and if A is false, A = ⦰, and A = . Therefore, whether A is true or false, A is not A.

Next, we may translate the logical statement “A or B” as AB. Proof: If A is true, and B is true, then A = and B = , and AB = .  By application of Equation (II), = , indicating that the statement AB = is true. Given AB, if one or the other, A or B is false, and the other is true, then AB = , or = , and = by Equation (II), and is true. Finally, if both A and B are false, AB = ⦰⦰ = ⦰, which is true. Thus, in every case, in the language of the CoDD, AB is equivalent to the logic statement A or B. (Note that in the language of the CoDD, AB is not AxB as in conventional algebra. We have not yet defined the fundamental operations of arithmetic in the language of the CoDD.)

The CoDD forms of all the other basic statements of logic presented in the table below can be derived from these two with applications of Equations (I) and (II). To save space and time here, I’ll leave these proofs to readers who might enjoy developing them. The table is provided as a handy reference for use in applying the CoDD.

 Statement CoDD Translation Comments Cancel X= ﬧm ﬧmﬧn   ⇔     , or  ⦰ Cancel or Recall X Equivalence ﬧnﬧn ⇔ ﬧn Condense or Expand Not A Aﬧ Negate A or B AB Either or A and B ABﬧ Combination A implies B AﬧB Implication ﬧ True Existential

Note that when m = n ≥ 0, the CoDD initial equations revert to two equations operationally equivalent to the CoI initial equations. n n = n   and         n n   =    . Also, if all distinctions, including the arguments A and B, are of the same dimensionality, the logic statements above are operationally equivalent to the CoI logic statements. It follows, therefore, that, if all distinctions in a CoDD calculation are of the same dimensionality, the subscripts can be dropped. If, on the other hand, there are distinctions of two or more different dimensionalities in a calculation, all dimensionalities must be indicated with subscripts and the rules 1.) through 4a.) in Installment 3 must be applied, since otherwise, the outcome may be incorrect.

The interpretation of the primary algebra for logic is analogous to the discussion in Appendix 2 of the Laws of Form, with important differences: n indicates a ‘true’ and ‘existing’ distinction in an n-dimensional domain. It then follows that, if, by a finite number of steps consisting of legitimate CoDD substitutions and transformations, the algebraic expression representing a hypothesis can be reduced to n, it will be shown to be both a true statement and existing as a perceptual and/or conceptual distinction.

One of the most important features of the CoDD is that it can be used to prove scientific hypotheses. Using the CoDD to prove this hypothesis has a number of benefits: it provides a demonstration of the utility of the CoDD to prove important hypotheses in both mathematics and physics. It will provide a logical tool for dimensional extrapolation (the operational movement from one dimensional domain to another), and it will allow us to begin to glimpse the logical structure underlying the linkage between space, time and consciousness.

Proof of the Hypothesis of Embedded Domains

# The expression produced in this manner from the hypothesis essentially says that an n-dimensional domain containing an (n-1)-dimensional domain implies, by the converse of equation (3.), an n-dimensional domain containing two (n-1)-dimensional domains. Then, by applying (3.) repeatedly,

n-1 n n-1 n-1 n n-1 n-1 n-1 ﬧ... n-1 n-1 n-1n
= nnnn by (3.) and (4.a) repeatedly
= n by (3.)

[Notation: Where one or more small distinction symbol is followed by a larger distinction symbol, the upper arm of the larger symbol should be thought of as extended over the contained domain or domains.]
By demonstrating that the hypothetical equation reduces to n using Equation (3.) and Equation (4.a), we have proved that n contains an infinite number of (n-1)-dimensional domains for all n ≥ 1.

With this CoDD proof, the hypothesis of embedded domains rises to the status of a theorem. Because of the importance of this theorem as one of the concepts supporting dimensional extrapolation, we designate it as a principle:

Dimensional Invariance Principle #1 (DIP#1): The Principle of Embedded Domains:

An n-dimensional domain, n, (n > 0), contains an infinite number of (n-1) -dimensional domains.

In this post, the discussion has been about distinctions of extent and dimensional domains. The CoDD is applicable to all aspects of reality, so in the next In the next post, I will discuss another type of distinctions: distinctions of content.

1. Thank you for this introduction Edward. I'm looking forward to the next piece!

2. 'If perception is relative, and complete independence is an illusion, is there never an objective reality completely independent of individual consciousness? Well, Ed, in my mystically-initiated opinion, there has to be, if my Ultimate Force's Fundamental Calculus, based on 'Y=X Squared plus One' is to be taken into account, and where your TDVP treatise also becomes fundamental to the One of the formula. Full details follow, if hopefully of further interest:

The Ultimate Force’s Fundamental Calculus – According to Brian! The following URL refers:

3. My view is that the "most fundamental objective reality" is a finite collection of individual consciousness. The existence, autonomy and distinctiveness of each such individual consciousness, or free-will agent as I prefer to call it, is independent of all the others; therefore, there is a certain absolute distinction, ontologically, that can be drawn.
On the other hand, said agents, despite being autonomous, interact and impact each other in such a way that the perception, even *self*-perception, of any single one, becomes relative to the impact made by all others.
In consequence, the *perceived* reality, which is the one that matters since it is the one against which the agents operate, is effectively what they collectively make of it (hence, relative).

[and "the ME" within me is one such free-will agent]

1. I have no argument with this concept of collective conscious reality. You'll see, I think, that the CoDD is meaningful and operational in this context.

2. I think this particular orientation matches particular well with renowed novelist Terry Pratchet and his Discworld universe. Where there are little gods and big gods all existing based purely on their believers willing them to exist.

4. Typo correction to my comment above: 'If perception is relative, and complete independence is an illusion, is there never an objective reality completely independent of individual consciousness? Well, Ed, in my mystically-initiated opinion, this has to be, if my Ultimate Force's Fundamental Calculus, based on 'Y=X Squared plus One' is to be taken into account, and where your TDVP treatise also becomes fundamental to the One of the formula. Full details follow, if hopefully of further interest:

The Ultimate Force’s Fundamental Calculus – According to Brian! The following URL refers:

5. I'm still going over the proof, but, there is an interesting thing to comment:

In computer programs, some systems make an explicit usage of something we call "triboolean logic", that is logic in which boolean expressions can be either true, false or indeterminate (or uncertain).
A computer program that uses that formalism looks like this:

if ( certainly(a < b) )

this is the case where ( a < b ) -> true

else if ( certainly_not( a < b ) )

this is the case where (a < b ) -> false

else

this is the case where is uncertain whether ( a < b ) is true or false

There are several domains within computational systems (computational geometry for instance) in which programs must explicitly use this logic.

Incidentally, I introduced this formalism in a couple of computer systems so I'm quite familiar with the usage of such multivalued-logic.

Some computer systems also formalize the "meaningless" option of ordinary boolean logic, but are much less common that the example I presented.

6. Thank you for this comment. In the preface to Laws of Form, George Spencer Brown said that he and his brother had been using the Boolean counterparts of i and a+bi in practical engineering for several years before realizing what they were. And this was written in 1972. But what I find to be important in the application of the CoDD, is how the appearance of i in an n-dimensional problem indicattes the existence of an additional dimension.

7. Oh, that's interesting! I thought this was rather new.
It is indeed fascinating how sometimes we discover purely mathematical or logical things, then we realize that they actually reflect a part of reality.

8. Something that both Dr. Neppe and I hold to be a basic axiom of TDVP is that reality is a meta-logical system and mathematics is a reflection of the structure of reality.