**REALITY
MATHEMATICS**

**INTRODUCTION**

The theme of this presentation is that real mathematics
is not a human invention. The language, and notation, the ways in which mathematics
is described, may be human inventions, but the mathematical patterns of logical
order observed as cause and effect, are not. They exist in nature. The universe
is obviously dynamic, and the changes we observe and experience in the universe
proceed in a well-ordered manner, and have apparently done so for billions of
years; if that were not the case, there would be no such thing as science, nor
even a universe. Our very existence depends on the continued orderly cycles of
our natural environment, and that continued progression of orderly cycles
depends upon stability at the very most basic level. That basic stability
starts with the structure of the proton, the most stable compound structure in
the universe. The breaking down, or radioactive decay observed in every other
compound sub-atomic or atomic structure, has never been observed to happen to a
proton. No one has seen a proton decay. If protons do decay, particle
physicists estimate the half-life of that decay would be longer than the
estimated big-bang age of the universe.

With the addition of an electron, balancing the
positive charge of the proton, the remarkable stability of the proton is passed
on to the hydrogen atom, and this simplest of the atoms becomes the most
abundant element in the universe. Gottfried Wilhelm Leibniz declared that the
question: “*Why is there something rather than nothing?” * is the first question science should seek to answer.
To rephrase that seminal question, seeing that the proton is the basis of
atomic stability, we must ask: Why are protons so stable? The answer turns out
to be quite simple: Quarks, the quantum constituents that comprise protons, are
not particles as physicists assume; they are energy vortices, spinning in
three, six and nine dimensions, and protons are remarkably stability because two
up-quarks and one down-quark are not just stuck together like tinker toys to
form a proton, they merge like drops of water when they combine. The truth of
this will be demonstrated in this presentation.

The main purpose of this presentation is to provide
the rational foundation for, and the basic development of a primary quantum calculus
capable of comprehensively representing the logical structure of reality. I
will approach this ambitious task in three steps: 1) by recounting a brief
overview of the history of the development of rational thought during the last 2,500
years or so, for the purpose of identifying the mathematical logic of the innate
structure of reality; 2) by focusing on the most important ideas, concepts and
theorems that point the way to a mathematics of reality that will re-unify
mathematics, symbolic logic and the natural sciences; and 3) by describing some
of the insights gained from the application of the integrative quantum calculus.
The key ideas of the focus in steps #1 and #2 will include GÓ§del’s
incompleteness theorem, G. Spencer Brown’s *Laws
of Form,* and the principles of relativity and quantum physics.

The historical separation of the study of the innate
logical structure of reality into separate intellectual disciplines like philosophy,
number theory, algebra, geometry, quantitative analysis, the infinitesimal
calculus, physics, chemistry, psychology, medicine, linguistics, anthropology, etc.,
etc. …,was necessary. It took a number of very dedicated, serious thinkers a
few hundred years to construct the conceptual bases and specialized languages for
each of these disciplines. Not even polymaths like Leibniz and Newton lived
long enough to re-integrate the various disciplines into a consistent
epistemology, and turn to the task of developing a reasonably complete
ontology, i.e., an understanding of the true nature of reality.

Specialization, while necessary at first, quickly became
a barrier to any comprehensive understanding of the holistic nature of reality.
Today, specialists in one field can barely communicate with specialists in
another field. Each discipline has its own language. For example, when I
switched from the study of pure mathematics and theoretical physics to engineering
in the middle of my academic career, to escape the ivory tower of academia, I
was amazed to find that engineers had re-invented the wheel, so to speak, by
developing their own languages, with special terminology and notations for basic
parameters and mathematical procedures that had already been more efficiently defined
in the natural sciences and pure mathematics. Specialization is modern
science’s Tower of Babel.

As it turns out, the separation of disciplines,
especially the separation of physics, mathematics and logic, into functionally
independent disciplines, is a mistake that has led to the conflation of
non-existent abstractions with reality, and logical consistency with truth. G.
Spencer Brown partially rectified some of the confusion caused by this artificial
separation of disciplines, and re-connected logic and mathematics, by
introducing complex values analogous to complex number variables (expressed
algebraically as a + bi) into the algebra of logic in his calculus of indications.
But he fell short of re-connecting primary calculus with physical reality, even
though he remarked on the similarity of the forms produced by the introduction
of imaginary numbers into logic to the laminated onion-like structure of atoms.
(*Laws of Form* p. 105) He also alluded to the role of consciousness in
the development of the calculus of indications, but did not attempt to
represent consciousness as a variable in the equations of the calculus, as we have
done.

Brown re-connected mathematics and logic with a
wonderfully functional interpretation of the calculus of indications for logic,
in *Laws of Form*, but then observed that determinations of the *truth*
or *falsity* of statements made in the calculus could not be related to *existence*
or *non-existence* without risking the creation of logical paradox. He
concluded that appealing to existence was, after all, not necessary in order to
solve problems within the calculus. In addition, Brown concluded that while
consistency and completeness could be demonstrated within the calculus, proof could
only be established by appealing to a more comprehensive logic existing outside
the calculus. This conclusion is consistent with Tarski’s theorem, and suggestive of GÓ§del’s incompleteness theorem.
The connections between these theorems in pure theoretical mathematics and the
natural laws of physical, mental and spiritual reality will be made clearer as
we develop the calculus of distinctions, the quantum mathematics of reality.

To start with, we need to spell out some definitions
in detail, so the reader does not have to speculate about any of the terms used,
or supply definitions of his or her own, that may or may not coincide with the
meaning intended.

**BASIC
DEFINITIONS:**

**Calculus**:
A calculus is a system of logic incorporating calculation. There is not just
one calculus, several can be identified, associated with varying types of *a priori* assumptions.

**Calculation**:
A calculation is a process that transforms symbolic representations of patterns
of conceptual and/or existential distinctions from one form to another,
different, but equivalent form. For example, 1 + 2 = 3 is an application of the
fundamental mathematical operation of addition that transforms one symbolic
representation, 1 + 2, to a different, but equivalent form symbolized by the
integer 3.

**Distinction**:
In the primary calculus of distinctions, a distinction is defined as anything
that can be distinguished from its surroundings in any definable or describable
way.

**Equivalence:**

In the Calculus of Dimensional Distinctions (CoDD), A
is equivalent to B, if and only if, they simplify to the same basic value of
the calculus.

**Simplification**:

A
complex CoDD expression can be simplified by substitution of an equivalent,
simpler expression. Examples will be provided later in this discussion.

**Substitution**:

The
replacement of one expression by another equivalent expression.

**Existing versus Conceptual Distinctions**

In
this discussion, because the calculus is anchored in reality, I am able to
equate the term ‘existing’ with true and
‘real’, as opposed to purely conceptual, signifying a mental image or
construction that has no actual counterpart in the real world. By ‘real’, I
mean either something existing physically, or something having measurable impact
on physical reality, or both.

**Consciousness**:
Consciousness is primary. That means that it cannot be defined in terms of
anything else, and the correct answer to the oft-asked question “What is
consciousness?” is: Consciousness is the *a
priori* existential reality through which all things are perceived. It is
that which gives rise to awareness, without which nothing could be said to exist.

**Dimension**:
It is necessary to define the term ‘dimension’ very precisely for use in the
calculus of distinctions, because currently, the term is used in a variety of
ways in various fields of human thought, and thus can mean different things to
different people. In the calculus of distinctions, it is limited to mean
extension, nothing more, and nothing les: *A dimension is measurable in units of a
variable of extent in space, time or consciousness. Other types of variables are not dimensions,
and should not be confused as such.*

**TRUE**: In the physical
universe, CoDD variables of space and time are quantized in multiples of the Triadic
Rotational Unit of Equivalence (TRUE), the quantum equivalence unit based on
the physical characteristics of the electron. Dimensional variables of extent
in consciousness related to mental images, are reflections and contractions or
expansions of the dimensional variables of extent in the physical universe.

**Domain**:
Like dimension, the term ‘domain’ means a variety of things in common parlance,
so we must also have a precise definition for the term domain in the calculus
of distinctions. *A domain is a region defined in terms of variables of extent*. *An
n-dimensional domain is a region defined in terms of n dimensions*.
Space-time, e.g., comprises a 4-dimensional domain symbolized by 3S-1t, a
volume is a 3-D domain, an area is a 2-D domain, a line is a 1-D domain, and
point is a zero-D domain, known as a mathematical singularity in conventional
mathematics.

**Variables of Content, Extent and
Intent**

C*ontent*,
i.e., energy and mass as energy equivalence, is quantized, and because of this,
space and time, or space-time *extent*,
which, as Einstein pointed out, is not fundamental because it has no existence
of its own, must be considered to be quantized as well. *Intent* implies consciousness, and must be included because it
impacts reality. The necessity of the quantization of space-time becomes very
clear with the application of the calculus to phenomena measured in multiples
of the quantum equivalence unit. Referring to a region of space-time smaller
than the volume of the quantum equivalence unit has no meaning in quantized
reality, because it leads to the confusion of particles and waves and a variety
of problems related to the Pauli exclusion principle and what is known in
contemporary quantum physics as ‘the measurement problem.’

This brings up some important points regarding the
calculus of distinctions. When dealing with distinctions of extent, the
calculus becomes the Calculus of Dimensional Distinctions (CoDD). In this
calculus, a mathematical singularity, zero and nothing should not to be
confused with each other. For the calculus to be properly related to the real
world, the difference between these three concepts must be clearly defined.

**The Concepts of a mathematical
singularity, Zero and Nothingness**

With the application of the calculus of distinctions
to the real world, we must be clear that the state of non-distinction is not to
be confused with the concepts of singularity, zero and/or nothingness. When
counting objects, like apples or oranges, zero denotes the absence of a
specific kind of object, not a state of nothingness. Similarly, the absence of
a distinction does not indicate a state of nothingness, it just means that
there are no distinctions in the field of awareness. The numerical value of
zero is also sometimes confused with the concept of nothing. Zero can be assigned
to a point on a line between other points representing positive and negative values.
But, in quantized reality, dimensionless points (mathematical singularities),
one-dimensional lines and 2-imensional planes do not exist; they are conceptual
mental constructs only, giving rise to the confusion that something can be
nothing, and that something might arise from nothing.

**Distinguishing Conceptual
Distinctions from Existential Distinctions**

In today’s mainstream science, applications of
macro-scale mathematical tools to quantum phenomena have led to serious
confusions of abstract conceptualizations with existential distinctions, i.e.,
real phenomena. The inappropriate application of Newtonian calculus is perhaps
the most obvious mistake of this sort, but there are even more basic instances
of this kind of confusion. For example, the concepts of mathematical
singularity (the point), line and area are often imagined to be structures that
exist in the real world, but, in fact, they are purely conceptual and cannot
exist in the quantized reality of the physical universe. Distinctions of less than
three dimensions have no volume, and thus no capacity for substantial content.
Replacing the concept of a point with one volumetric quantum equivalence unit
(the TRUE) in the calculus of distinctions, and thereby accurately reflecting
physical reality, allows us to clarify the difference between conceptual mental
images and distinctions that exist in the real world.

**Hilbert Space and CoDD n-Dimensional
Domains**

In
previous publications, I have used the term Hilbert space, named after David
Hilbert, the mathematician who extended Descartes’ 3-D space in which objects
are located by three numerical coordinates, to n-dimensional space to
accommodate non-Euclidean geometries. There are, however, significant
differences between the classical concept of Hilbert space and the
n-dimensional domains of the calculus of distinctions. In the CoDD, dimensions
are limited to aspects of reality that can be measured in integral variables of
extent. In the CoDD, Mass, energy, density, and other quantifiable features of
objects existing in an n-dimensional domain, are measured using variables of
content and/or combinations of variables of content, extent and intent, and
should not be confused with dimensions, which are measured in variables of
extent. To avoid this kind of confusion in the CoDD, dimensions are strictly
limited to realities measured in variables of extent, and those variables are
defined in terms of quantum equivalence units. In addition, in the CoDD domain,
roughly analogous to Hilbert space, the dimensionless point is replaced by the multi-dimensional
volume of the Triadic Rotational Unit of Equivalence.

**A Triad of Variables are needed to
Describe Existential Distinctions**

Mass,
energy, density, and other features of content in the quantized world, are
measured using variables of content and/or combinations of variables of
content, extent and intent. An existential object is only fully described by
including variables of extent, content and intent or informational meaning
associated with measurable impact. The three values possible in the CoDD are:
1) True, or real; 2) False, or non-existent; and 3) imaginary (analogous to a +
bi), or extra-dimensional. With these concepts and definitions, as outlined
above in mind, we can develop a useful functional notation for the calculus of
dimensional distinctions.

**(To be Continued)**