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**THE MATHEMATICS AND GEOMETRY OF CONSCIOUSNESS**

The goal of natural science from the beginning has
been to explain everything. In modern times, this quest has been articulated as
the search for a ‘theory of everything’ abbreviated as TOE. Einstein’s unsuccessful
quest for a unified field theory, a theory combining all the physical forces in
the universe in one consistent theory, has been interpreted by physicists since
Einstein as a quest for a TOE, reflecting the belief that everything can be
explained in terms of physical principles. For twenty years I have been saying
to anyone who would listen that there can be no real TOE unless consciousness
is included in the equations describing reality. See “Transcendental Physics”,
page 208. I am not alone in this, others, like Peter Russell, Amit Goswami, Sir
Roger Penrose, Stuart Hammeroff, and Vernon Neppe, to name a few, have been
saying something similar. But mainstream science has not been able to define
what consciousness is, let alone represent it in the equations of a TOE.

To understand

*why*modern science has not been able to put consciousness into the equations describing the laws governing the physical world, and why modern science has found no way even to define consciousness in relation to physical reality, we must go to the roots of the axiomatic approach used in modern mathematics. We must go back nearly 2,600 years to a period of about 370 years in length, to the world of Pythagoras (582 -507 BC), Plato (428 -348 BC), Euclid (325 -265 BC), and Archimedes (287 -212 BC), and review the ideas of these ancient Greek natural philosophers, because they formalized the ideas that make up the foundations of the modern understanding of mathematics and geometry, because, as I will explain, consciousness, mathematics and geometry are intimately related.
The axiomatic approach, developed by the Greeks,
starts with the definition of a set of self-evident facts (axioms) and then
derives or deduces logical conclusions that must be true if the axioms are
true. Pythagoras used the axiomatic/deductive method to prove his famous
theorems. Plato regarded axioms as reflections of a ‘perfect’ reality of which
matter and energy were only imperfect reflections. Euclid formalized the axiomatic approach
applied to geometry in his ‘Elements’ of Geometry. And Archimedes applied the
axiomatic approach to practical problems, and became the first engineer and
experimental scientist in the modern sense. The difference between Archimedes’
pragmatism and Plato’s idealism is roughly the same as the difference between
the experimental and theoretical scientists today. I believe that we need both
Platonic and Archimedean scientists, but in this discussion, I intend to show
how the predominance of the Archimedean approach in modern science has led to a
misunderstanding of what mathematics, geometry and consciousness actually are. And
we must go back to Euclid to see where and how the thinking deviated from the
path that leads to defining consciousness.

Using the axiomatic approach that emerged from the
thinking of the three Greek philosophers, Pythagoras, Plato and Euclid, as
pragmatically interpreted by Archimedes, early modern scientists, notably
Descartes and Laplace, diverted scientific thought into the dualistic
interpretation of reality that led to the reductionist materialistic philosophy
of science prevalent today, reflected in the ‘Standard Model’ of physics.
Reductionist materialism leads naturally to a belief in absolute determinism as
reflected in Descartes and Laplace’s statements in the 1700’s to the effect that
it would only take a few years for scientists to determine the initial
conditions of the universe, after which the complete history and fate of the
universe could be calculated using Newtonian mechanics. A more recent statement
of belief in determinism is found in Stephen Hawking’s “A Brief History of
Time’, 1988. He predicted a TOE by the year 2000. …Of course that didn’t happen,
because, as I will explain, consciousness has been left out of the equations.

In the reductionist worldview of modern science,
geometry and consciousness have one thing in common: they are both relegated to
non-substantive roles in the universe, related only secondarily to the dynamics
of matter and energy. In modern mainstream science, geometry is seen as the
description of space-time, a passive backdrop to the dynamic interactions of
matter and energy. And consciousness is seen as an emergent feature of matter
and energy at certain, as yet, not well-defined levels of complexity. In this
view, the geometric features of space-time are shaped by random variations of
mass and energy throughout the universe; and consciousness is seen as an
emergent, developing awareness, appearing only in organic life forms.

On the other hand, the theories of relativity and quantum
mechanics, verified many times over by empirical data, suggest in different
ways, that these conceptualizations of consciousness and geometry are flawed,
and if not completely incorrect, at the very least, incomplete. Relativity, for
example, reveals that the measurable features of physical reality depend upon
the location and velocity of the observer relative to the objects of
observation, and quantum mechanics tells us that the physical form exhibited by
a quantum system depends upon choices made by a conscious observer. In both
cases, the reality we can observe, measure or in any way experience is affected
by the conscious observer. Based on these clues, is it possible that mainstream
science has it backward? Could it be that instead of being secondary and
emergent, mathematics, geometry and consciousness are actually fundamental
aspects of existential reality?

The findings of TDVP support this idea, and Close and
Neppe are not the only ones finding this. Books by Sir Roger Penrose and the
research of Penrose and Hammeroff, Peter Russell, Amit Goswami, and a growing
number of researchers support this view. What sets The Neppe-Close theory apart
from the others is the fact that while others talk about the need to put
consciousness into the equations of science, we have actually done it. As an
important result of doing so, we have been able to explain things not explained
in the current paradigm, things like why quarks combine in threes, the
intrinsic spin of fermions, and other things that have puzzled scientists and
mathematicians for half a century or more.

I have suggested that it is the Archimedean
interpretation of the axiomatic method of Pythagoras, Plato and Euclid that has
blinded mainstream scientists to the reality of consciousness, the reality of geometry
and their intimate relationship. Let me explain this in a little more detail. Here
are Euclid’s five axiomatic (self-evident) statements as translated from the
Greek:

1. It is possible to draw a straight line from any point to another point.2. It is possible to draw a finite straight line continuously in a straight line.

3. It is possible to draw a circle with any center and radius.

4. All right angles are equal to one another.

5. If a straight line drawn across two straight lines forms interior angles on the same side less than two right angles, the drawn lines will meet somewhere on the side on which the angles which are less than two right angles lie.

These axioms are a mixture of platonic and pragmatic
interpretations of geometric features of reality. I said pragmatic, rather than
Archimedean because when Euclid wrote his Elements, Archimedes hadn't been born
yet. Archimedes focused on the pragmatic aspects of the Elements, as mainstream
scientists have ever since. Let me explain how these statements are reflective
of both Platonic ideals and practical application. The statements as written
focus on the practicality of the physical representation of points, lines and
angles using simple drawing instruments. The constructions presented in
Euclid’s Elements are achievable using only compass, pencil and straightedge. The
features of space that underlie these statements that are considered to be
self-evident, can be seen more clearly if the statements are rewritten as
follows:

1. A straight line is defined by the shortest distance between two points. 2. A finite straight line is continuous between any two points, and can be extended as far as we like.

3. Space is such that circles of any size can be constructed around any given point.

4. All right angles are equal to one another. (A right angle is defined as exactly one-fourth of a circle, and when super-imposed, all right angles are exactly congruent.)

5. If one straight line crossing two straight lines forms interior angles less than right angles on one side of the line, the two straight lines will meet at some distance away on that side of the crossing line. Visualizing this we could easily add that if the two angles are larger than right angles, the two lines will meet some distance away on the other side of the crossing line, and if the interior angles are both right angles, the two lines are parallel, and will never meet.

Note that in this idealized (Platonic) form of
expression of Euclid’s axioms, points have no dimensional extent, lines are
one-dimensional, with no thickness, and circles lie on a two-dimensional plane.
In application, i.e. in practical construction with drawing instruments, however,
they all have three dimensions, just as everything in the material world has.
The representation of points, lines and angles may be a smudges of graphite on
a sheet of paper or indentations drawn with a stick on a smooth area of sand.
In either case, the representations are three dimensional. We can conceive of
dimensionless points, and one-dimensional lines and two-dimensional plane
surfaces, but in material representation, or as they relate to objects in the
physical universe, they are three-dimensional. In fact, the elements of quantized
existential reality are necessarily at least three dimensional.

This can be seen clearly as follows: A dimensionless
point has no extent, therefore it cannot contain anything substantial. A one-dimensional
line contains an infinite number of dimensionless points, but has no thickness,
so it also cannot have existential content. Similarly, a two-dimensional plane
containing an infinite number of one-dimensional lines, and a doubly infinite
number of dimensionless points, has no capacity for existential content. Only
geometrical forms of three dimensions or more are capable of containing
existential substance.

This disparity between idealized conceptualization within
our minds, and the world ‘out there’ that we experience in a limited way
through our senses, must be borne in mind when applying mathematical and
geometrical concepts to form any model of reality. This is the root of the confusion
that causes scientists to think that there is one set of rules for the macro
scale universe, another for the quantum scale, and that the two are
incompatible. This is a confusion arising from the illusion that the internal
and external worlds of our experience are separate worlds, and that the quantum
realm, our everyday world, and the expanding cosmological universe are separate
realities. They are not. There is only one reality. As Erwin SchrÓ§dinger said
in his book ‘What is Life?’, “The world is given to me only once, not one
existing and one perceived.”

We need to be very clear about this because it is the
cause of much confusion. A TOE is a model. It is a model based on mental
concepts existing in someone’s imagination, stored in their brain. But the
brain, concepts, and model represented by language and mathematical symbols,
are all part of the same reality. The model, moreover, must not be mistaken for
the reality. We would never mistake a map, however detailed, for the
countryside it represents, and we must never mistake our models of reality,
which are based on the incomplete information obtained through the senses and
processed in the brain, for the reality we are trying to represent with them.

Classical physics provides a pretty good model of the
part of reality experienced on the scale of the physical body and sense organs.
This happens to be the midrange of reality. With the refinements of special and
general relativity, the model provides a pretty good map of observable reality
on the cosmic scale. And quantum mechanics provides a working model of reality indirectly
detectable on the quantum scale. If the models don’t agree in the areas where
they overlap, it doesn’t mean that reality operates by different rules at
different scales, as scientists like to say it does, it means the models are
wrong. Reality has no inconsistencies in it, the inconsistencies are in our
models.

However, it would be a mistake to say the models are completely wrong. They are not, they are only demonstrably wrong in the areas where they disagree. Like in the case of classical physics and relativity, it is probable that the models are incomplete in a way that makes them inaccurate beyond the scale in which they were conceptualized. The point here is that the so-called ‘Standard Model’ is actually a hodge-podge of models that fit together loosely and imperfectly, with conflicts and some holes that are not addressed at all. But this is not a bad thing. In fact, it is a good thing because it tells us that we need to go back and look at the axioms upon which these models were built, and the mathematical tools that were used to build them.

However, it would be a mistake to say the models are completely wrong. They are not, they are only demonstrably wrong in the areas where they disagree. Like in the case of classical physics and relativity, it is probable that the models are incomplete in a way that makes them inaccurate beyond the scale in which they were conceptualized. The point here is that the so-called ‘Standard Model’ is actually a hodge-podge of models that fit together loosely and imperfectly, with conflicts and some holes that are not addressed at all. But this is not a bad thing. In fact, it is a good thing because it tells us that we need to go back and look at the axioms upon which these models were built, and the mathematical tools that were used to build them.

Returning to geometry and our representations of it, Platonic
points, lines and angles have zero, one and two dimensions, respectively, but
points, lines and angles as we experience them in reality, are all three
dimensional. This tells us that if we want to understand what geometry is, we
need to re-align the axioms of Euclid and our mathematical tools with quantum reality.

The realignment needed to construct a comprehensive model of reality goes beyond just recognizing that our experiences of geometrical realities are three dimensional, it requires realizing that each and every finite distinction that makes up our experience of reality is at least three dimensional. This is, in fact, the real, most important message of quantum physics. The substance of reality, measurable in units of mass and energy, is quantized, and our experience of it is always in multiples of quantum units. Because of this, a complete overhaul of the current mathematical/logical system used to describe reality is needed.

Newtonian calculus is a wonderful tool to describe motion in three dimensions of space, one dimension of time

The realignment needed to construct a comprehensive model of reality goes beyond just recognizing that our experiences of geometrical realities are three dimensional, it requires realizing that each and every finite distinction that makes up our experience of reality is at least three dimensional. This is, in fact, the real, most important message of quantum physics. The substance of reality, measurable in units of mass and energy, is quantized, and our experience of it is always in multiples of quantum units. Because of this, a complete overhaul of the current mathematical/logical system used to describe reality is needed.

Newtonian calculus is a wonderful tool to describe motion in three dimensions of space, one dimension of time

*in mid-scale reality*, however, it brings with it axiomatic assumptions that are invalid for describing quantum reality, and that actually*causes*conflicts between the Classical/Relativistic-scale model and the quantum-scale model. Newtonian calculus assumes that space and time, the measurable geometrical variables of reality are continuous, implying that they are infinitely divisible. This leads to results implying that substances (mass and energy) are also infinitely divisible, which is not true in a quantized reality.
Since the substance of reality is quantized, not only
are points, lines and angles three dimensional, but any geometrical structure
forming the boundaries of a distinction consisting of one or more quanta of the
substances of reality must also be at least three dimensional. So the geometry
of existential objects is necessarily existentially quantifiable. This means
that the results of applications of Newtonian calculus at the quantum scale are
inaccurate, and in some cases, existentially incorrect. Newtonian calculus
works at the midscale because the quanta of reality are so infinitesimally
small relative to midscale observations and measurements that the errors in
calculated results are undetectable. But at the quantum scale, these errors are
catastrophic.

To see how our understanding of geometry must change
because of the discovery that reality is quantized, and how our mathematical system
of logic must change, we must go back to Euclid’s axioms. To develop a
mathematical system designed for application to quantized reality, we must
define the mathematical elements of geometry in existential, not idealized
terms. The concept of an existential point is the basic concept upon which the
new geometrical mathematics must be built. An existential point in a quantized
reality is a three- or more-dimensional minimal quantum volume, not an
idealized dimensionless singularity.

I first conceived of, and began to develop the
appropriate geometrical mathematics for application to quantized reality, the
calculus of distinctions (CoD), in 1986. The basic logic of the CoD was
published in my second book, “Infinite Continuity’ in 1990. The derivation and
further development of the CoD into an effective dimensionometric mathematical
system has been published more recently in “Reality Begins with Consciousness”
(Neppe & Close, 2011, www.BrainVoyage.com)
and a number of technical papers in professional journals. These derivations
are beyond the scope of this discussion, but can be described here in general
terms.

As a system of symbolic logic, the CoD has its roots
in conventional Boolean algebra and George Spencer Brown’s calculus of
indications (“Laws of Form”, George Allen and Unwin, London, 1969). However, the
CoD is fundamentally different from these logical systems in three important
ways: First, it incorporates axiomatic geometry (dimensionality) into its
notation. Second, the basic unit of distinction is the existential three-dimensional
quantum point. In traditional systems of symbolic logic and Brown’s Laws of
Form calculus of indications, unitary existence is neither essential, nor
necessary for application to problems of mathematics and logic, but existence

*is*a requirement for the basic unit of a mathematical system of logic designed to apply to an existential quantized reality. Third, the basic existential quantitative unitary distinction of the CoD is derived from the empirical data of the Large Hadron Collider (LHC), relating it solidly to the reality we experience through the physical senses.
The derivation of the basic existential quantitative
unit of the CoD, which I call the Triadic Rotational Unit of Equivalence or
TRUE quantum unit, from first principles of relativity and quantum mechanics, is
beyond the scope of this post, but has also been published in the references
cited above. TRUE units are derived through a method of normalization similar
to the way Planck units are derived, but differ significantly from Planck units
because Planck units are normalized to five universal constants, while TRUE
units are normalized to the mass of the electron. It is important to note that TRUE
units are derived from empirical data, and that the derivation and definition
of the TRUE quantum equivalence unit from empirical data and the principles of
relativity and quantum mechanics as the unitary quantum distinction of the Cod,
allows us to avoid the inaccuracy and errors of the application of Newtonian
calculus to quantum phenomena. It also allows us to start our CoD analysis at
three dimensions, the point where conventional mathematics becomes very
difficult and often intractable. The TRUE quantum unit also integrates
relativity and quantum mechanics by providing unitary equivalence of mass,
energy, space, and time as experienced by conscious observers drawing
meaningful distinctions in the quantized images of reality delivered to their
conscious awareness by the physical senses.

Clearly, proof of the equivalence of all of the existential
parameters of measurement defining the minimum finite unit of distinction, the
Triadic Rotational Unit of Equivalence, is crucial to this model; and just as
clearly, that proof is very complex and subtle. It is also admittedly controversial,
because it relies on defining the first existential distinction as the
conscious distinction of self from other. But any trepidation we may have had regarding
the validity of this approach, was dissipated by the fact that it allowed us to
bring consciousness into the equations of science in a very real and meaningful
way, which further resulted in a rational explanation of why the dynamically
spinning structures we call electrons, protons, neutrons atoms and molecules are
symmetrically stable, allowing them to exist long enough to support life as we
know it.

The power of the CoD is yet to be fully realized. So
far I have used it to streamline logical analyses, giving rise to proofs of
several important scientific hypotheses and mathematical theorems, and the
development of several new mathematical procedures, including Dimensional
Extrapolation, the unitary projection from any n-dimensional domain into the
n+1 dimensional domain, and derivation of the multi-dimensional quantum
Diophantine Conveyance Equation. (A ‘Diophantine’ equation is simply an
equation that is satisfied by integer solutions. It should be clear that with
application to integer multiples of the TRUE unit, solutions of Diophantine
equations are appropriate and necessary for use in models describing quantized
reality.)

The Conveyance Equation expresses the logical
structures of hyper-dimensional domains as they are conveyed mathematically
into one- two- and three-dimensional domains. These logical structures include:
the fundamental operations of integer arithmetic in the 1-D domain, the
Pythagorean Theorem in the 2-D domain, and Fermat’s Last Theorem in the 3D
domain. Application of these three specific subsets of the Conveyance Equation allows
us to explain why quarks only occur in triadic combinations, and applications of
these subsets of the Conveyance Equation using TRUE units, allow us to develop
analyses of electrons, protons and neutrons explaining why they are stable, and
explaining why certain stable natural elements, e.g., Carbon, Hydrogen, Oxygen,
Nitrogen, etc. form organic life through which consciousness is manifested in
the physical universe.

Application of the CoD and TRUE quantum unit analysis has
also revealed the existence of a third form of the substance of existential
quantum reality, besides mass and energy, that is necessary for there to be any
stable structure in the physical universe. Dr. Neppe and I decided to call this
third form ‘gimmel’ to distinguish it from mass, energy, space and time
parameters. Gimmel is present in specific numbers of TRUE units just as mass
and energy are, and the exact number of TRUE units of mass, energy and gimmel
in each of the elementary particles making up the elements of the Periodic Table
is directly determined from empirical data and well-established physical
principles.

It is important to note that gimmel has been and

It is important to note that gimmel has been and

*is*instrumental in conveying logical structure into the 4D domain of space-time. We hypothesize that this logical structure is a form of consciousness conveyed from the transfinite and infinite domains of hyper-dimensionality into the 4D domain that we experience through the senses.
The most revolutionary finding of TDVP is the finding
that the existence of the universe in any, even semi-stable form depends upon the
existence of gimmel before, during and after any origin event giving rise to
the physical universe as it exists now. The undeniable interdependent existence
of TRUE units of mass, energy and gimmel means that, while organic life is undoubtedly
emerging from physical evolution, as current mainstream science contends, some
form of gimmel, as the carrier of logic, meaning and consciousness has always
existed, otherwise, nothing could exist because there would never have been
anything from which it could evolve.

The fact that the existence of gimmel is necessary for any long-term structural stability in the physical universe, and the fact that its presence in structures of matter and energy provides logical consistency and meaning not found in purely random processes, suggests that individualized consciousness, manifest in finite organic life forms, is orchestrated by gimmel to impact physical reality like points of light shining through the filter of the mass and energy of particles and waves and the transfinite domains of hyper-dimensionality, emanating from an infinite source beyond space-time.

The fact that the existence of gimmel is necessary for any long-term structural stability in the physical universe, and the fact that its presence in structures of matter and energy provides logical consistency and meaning not found in purely random processes, suggests that individualized consciousness, manifest in finite organic life forms, is orchestrated by gimmel to impact physical reality like points of light shining through the filter of the mass and energy of particles and waves and the transfinite domains of hyper-dimensionality, emanating from an infinite source beyond space-time.

The Code within the Seed of the Universe Itself, the Mind of the Ultimate Force, the 'Triadic One' of the mystically-inspired formula 'Y = X squared plus One' - One problem remains, though, Ed: What is to be the 'fourth ingredient' of this formula that will, of necessity, balance ( cosmicate) and thus prevent humanity from destroying the planet and itself on route to a state of conscious perfection, ad infinitum? God (Ultimate Force) help, not only the USA, but us All!

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