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 ﬧ 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
Letﬧ indicate
the drawing of a distinction, and let A ﬧ 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; ‘ﬧ ‘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:
ﬧn specifies a distinction of n
dimensions, where n is an integer ≥ 0. Now, if ﬧn represents a
distinction of extent, as opposed to a distinction of content, then the symbolﬧ0
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.)
ﬧmﬧn = ﬧnﬧm =ﬧn,
if and only if n = m ≥ 0
3.a) If n > m, ﬧmﬧn = ﬧm + ﬧn = ﬧmﬧnﬧn. (Note
that ﬧmﬧn is analogous to ╗in
the CoI notation.)
3.b) If m > n, ﬧmﬧn = ﬧm + ﬧn = ﬧmﬧnﬧm
4.) ﬧmﬧn = ,
or ⦰ if m = n
4.a) ﬧmﬧn = ﬧn,
if n > m ≥ 0
Also, ﬧmﬧn 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, ﬧ
represents a distinction of extent and, in general, ﬧn represents an
n-dimensional distinction. This means that ﬧ0
is a dimensional singularity or point, ﬧ1,
a line, ﬧ2, a plane, and ﬧ3, a
volume. Thus ﬧn
represents an n-dimensional domain, capable of containing an infinite number of
distinctions of extent. For example, a three-dimensional domain, ﬧ3, can contain an infinite number of ﬧ2 domains, ﬧ2
can contain an infinite number of ﬧ1 domains,
and ﬧ1 can contain ﬧ0 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, ﬧn, 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 ﬧmﬧn indicates
that ﬧm is
nested in ﬧn, and ﬧﬧ ﬧ indicates that ﬧ and ﬧ are nested in ﬧ.
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:
ﬧmﬧn ⇔ . (I)
and ﬧnﬧn ⇔ ﬧn
(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 = ﬧn 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 ﬧ, through the
application of the initial equations, B is also true, and if the statement B
reduces to ⦰, implying that the
opposite of ﬧ 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 ﬧ. Proof:
If A is true, A = ﬧ, and by application of Equation (I), A ﬧ= ⦰, and if A is
false, A = ⦰, and Aﬧ = ﬧ. Therefore, whether A is true or false, Aﬧ 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 = ﬧ and B
= ﬧ, and AB = ﬧ ﬧ. By application
of Equation (II), ﬧ ﬧ =
ﬧ, indicating that the
statement AB = ﬧ ﬧ is
true. Given AB, if one or the other, A or B is false, and the other is true,
then AB = ⦰ ﬧ, or ﬧ ⦰ =
ﬧ, and ﬧ =
ﬧ ﬧ 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. ﬧn ﬧn = ﬧn
and ﬧn ﬧn = .
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: ﬧn 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 ﬧn,
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
We write the hypothesis of embedded
domains in terms of CoDD logical notation, as follows:
ﬧn-1ﬧnﬧ ﬧn-1ﬧn-1ﬧn
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,
ﬧn-1 ﬧn ﬧ ﬧn-1 ﬧn-1 ﬧn ﬧ ﬧn-1 ﬧn-1 ﬧn-1ﬧ ﬧ... ﬧn-1 ﬧn-1 ﬧn-1 …ﬧn
= ﬧnﬧnﬧn … ﬧn by (3.) and (4.a)
repeatedly
= ﬧn 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 ﬧn using Equation (3.) and Equation (4.a), we have
proved that ﬧn 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, ﬧn, (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.
