THE CALCULUS OF DISTINCTIONS
TABLE OF CONTENTS

Calculus of Distinctions
Non-Numerical Arithmetic
The Algebra of Distinctions
Types of Dinstinctions
Logical Interpretations of the Calculus

The calculus of distinctions, its arithmetic, and its algebra are clearly applicable to any type of distinction. In order to apply the calculus properly, however, we must carefully define the nature of the distinction being considered. The fact that we exist as complex beings, dealing both consciously and involuntarily with a variety of distinctions, makes it difficult for us to recognize the types of distinctions we are capable of making. We must analyze the differences inherent in the different types of perceptual distinctions and their implications in regard to understanding the nature of reality. We may start unraveling the tapestry of perception and inference by discussing some of the types of distinctions.

The first distinction that we must consider is the distinction of self from other-than-self (other).

Whether it is arbitrarily chosen or forced upon us makes no difference, for the purposes of the present discussion. This primary distinction, the distinction  between self and everything else, is a basic function of conscious being.

Self is generally thought of as inner, while other is conceived as outer. Self is capable of holding conceptual images, while other must have the potential to exhibit substance, extent and duration in order to be perceived by self as other. Hence, to individualized consciousness, all the properties distinguished in the physical universe exist only in relation to the primary distinction: self from other. Without this primary distinction, awareness of any other distinction lacks context.

The conception of inner and outer requires the drawing of a three-dimensional distinction. Outer represents an expansion of substance in all directions. Inner represents the contraction of substance toward a point. Because of the natural symmetry of the void, any distinction in the outer sphere must be three-dimensional in order to be existential. Distinctions of one or two dimensions may be seen, in every case, to be abstractions. Like the reflection of an object in a mirror, they are only representations of an aspect of the complete reality being distinguished. If a distinction has extent in less than three dimensions, it lacks volume and therefore has no substance. It is non-existent and discontinuous in the missing dimension. And so, in accordance with Consequence C of the basic assumption of infinite continuity, we must conclude that:

All existential distinctions are at least three-dimensional.

Therefore, in the application of the calculus of distinctions, we must consider the primary distinction and all distinctions of the non-numerical arithmetic and algebra of distinctions to be three-dimensional. This will advance the application of the calculus and give it a power unknown to conventional mathematics:

All operations of the calculus may be applied directly to three-dimensional phenomena, using expressions and equations of only the first degree.

The basic distinctions of time and space in other   cannot exist independent of the distinctions in substance. Just as the existence of other can only be known as distinct from self, the distinctions of space and time arise as the necessary media for making distinctions in substance (consciousness, mass and energy). Classical physical science has assumed that time, space, matter and energy are independent realities in and of themselves, but this cannot be true in an infinitely continuous universe. They are all interdependent as distinctions in other and ultimately owe their perceptual form and perhaps even their existence to the primary distinction of self from other.

Matter and energy are at opposite ends of the spectrum of physical substance and are, as science now knows, trans-formable from one to the other. They are simply different forms of the same substance. Matter is condensed energy, energy is expanded matter. Time and space are also dependent upon the distinctions in substance and are related to each other through the perceived transformations of matter to energy and energy to matter. As stated in the last section, the interaction of the three forms of substance creates the relative measures of time and space.

In order to analyze the process of perception and awareness, or the making of distinctions, we may visualize the primary distinction of the physical universe as a contraction or condensation in an infinitely continuous expanse of substance.

Within the conceptualizations of extent, we may visualize: zero-dimensional distinctions, or points in space; one-dimensional distinctions, or segments of a line; two-dimensional distinctions, or planar areas; three-dimensional distinctions, representing volumes; and four-dimensional distinctions, which are changes of three-dimensional forms over time. We must realize, however, that zero-, one- and two-dimensional distinctions are incomplete conceptualizations by themselves and exist only as mental images abstracted from three-dimensional distinctions.

Conventional mathematics bases all forms of analysis upon one-dimensional units, but for application of the calculus of distinctions, based as it is upon the assumption of an infinitely-continuous-self-referential reality, we must take into consideration the dimensionality of the distinction we choose to deal with before the step into numerical analysis can be properly taken. To clarify, first consider the fundamental operation of numerical analysis upon which the other operations of numerical mathematics are based. This is the operation of combination, which we call addition.

For zero-dimensional distinctions:

+ = Ex. 13.

Since points are dimensionless and any number added together are still dimensionless.

For one-dimensional distinctions:

+ = Ex. 14.

At this point, we have moved beyond the logic of the calculus of distinctions previously developed. We have introduced enumeration and equivalence in the form of separate and equal distinctions.

The form of the distinctions in Ex. 13 is the abstract dimensionless point. The form in Ex. 14 is a one-dimensional finite unit line segment. For two-dimensional distinctions, the elementary form is a unit area. While the addition of distinctions for one-dimensional units follows the rules of numerical arithmetic we are use to, addition for two-dimensional units does not. For example:

If represents a one-dimensional distinction,

If is two-dimensional,

Only in special cases does the addition of whole unit two-dimensional distinctions yield whole unit two-dimensional results. We know these as integral solutions of the Pythagorean Theorem equation. Examples:

For three-dimensional or existential distinctions, the elementary form is a unit volume. In this case, no combination of unit volumes can yield a volume of whole unit dimensions. This is established by the proof of Fermat's Last Theorem for the case n  = 3.

The symbols of numerical distinction are used to represent what we perceive in the outer world. Difficulties in understanding the nature of reality arise because of the use of inappropriate conceptual distinctions and the confusion of symbols with reality. For example: the numbers we use as the basis of mathematical physics are one-dimensional. We describe integers, fractions, etc., i.e. rational numbers, as lying on a one-dimensional number line. This line is infinitely continuous. The distinctions we perceive in physical reality, however, are of three or more dimensions and exhibit apparent discontinuity.

As we have demonstrated above, the zero-dimensional distinction is of no use to us for quantitative analysis. However, the one-dimensional distinction is. We may define a basic unit, as long or short as we like, and use it iteratively to describe distance along the line. The line is conceivably of infinite length and infinitely continuous, at least down to the smallest unit we can define. If we allow our basic units to have more than one dimension, traditional quantitative analysis becomes more complicated. On the other hand, if we stick with one-dimensional units as our basis, we must deal with equations of quadratic or higher degree when we try to use them to describe entities and events in the real world.

While one- and two-dimensional distinctions are of use in gaining an understanding of the use of the calculus, we wish to go quickly to the application of the calculus to a three-dimensional reality. By so doing, we will come into physics at a rather advanced stage.

The micro-scale horizon beyond which no smaller direct measurement is possible.

Imagine the smallest point detectable by whatever means, the eye or any extension of it. At this point, perceptually, anything smaller is undetectable, anything larger, if it exists as a physical object, has three dimensions. We will define this smallest detectable object as existing at what we shall call our Lower Limit Horizon (LLH). This is the vanishing point of =x in differential and integral calculus.

THE ORIGIN OF ASYMMETRY

The significance of our choice of the form of basic units of distinction may now be understood as follows: atomic and quantum theory are based upon the assumption that all substance is composed of basic structural units. Let's assume these units are also the basic units of distinction. Further, let's assume that they exist at the LLH and that they are symmetrical, like spheres or cubes. If two such symmetrical units combine, in a way that does not change their total mass or volume (such as sharing positive and negative charges) then by virtue of the proof of Fermat's Last Theorem for n= 3,  the new entity formed in this manner cannot be symmetric. Thus, a measure of complexity enters the universe based upon the simplest of distinctions at a very early stage. The inescapable asymmetry of the first particle made by combining elementary unit particles would affect the movement of particles under the influence of any uniform force. The new combined particle would wobble or spiral.

This conclusion concerning the asymmetry of secondary particles is reached by realizing that the basic units of distinction are three-dimensional. If is defined as representing the distinction of a symmetrical unit of three dimensions, then:

+ = , etc.

In general, if all distinctions are three-dimensional, then:

+ = Ex. 15

where a, b and c represent volumetric units. Translated into the one-dimensional number system we are accustomed to, Ex. 15 becomes:

Ex. 16

where x, y, and z

are one-dimensional units and the distinctions are spherical.

Canceling the common factor in Ex. 15 yields:
Ex. 17.

By Fermat's Last Theorem, which is easily proved for the case n = 3, no solutions are possible for Ex. 17 with x, y, and z whole numbers. Therefore, if  x and y are unitary volumes, z cannot be.

By visualizing the symmetrical contraction of some uniform substance, we have introduced all of the basic elements of physics: Matter and energy -- the substance; space -- the spherical extent of contraction and expansion; time -- the measure of change as the form (a symmetric spherical wave front) is transmitted through the space it defines.

But our model of reality goes beyond classical and modern physics, it includes consciousness as the self-referential principle that makes these distinctions possible. We are able, within this framework, to describe not only the physical universe, but also the way its perception arises from and relates to consciousness.

In regard to our own individual consciousness, we know that we may create three-dimensional existential distinctions as when, for example, we scoop a snow ball from a snow drift, or model something from a block of clay. Or, we may recognize what we perceive as existing distinctions, such as night from day, hot from cold, dark from light, etc. Notice, however, that all the distinctions entering our consciousness are relative. That is, they exist only in relation to the finite features or limits of our apparatus of perception. In this sense, at least, all our knowledge of reality is relative.

In an infinitely continuous universe, the forms of the various distinctions of our perception are determined by the physical limitations and the relative size of the apparatus of perception. If the assumption of infinite continuity is valid, then the assumptions of absolute separation and the acceptance of universal constants, such as Planck's constant and the constant speed of light as existential rather than perceptual, should lead to logical paradox. In relativity and quantum mechanics we do, indeed, find such logical contradictions. Examples are: the clock paradox and the Einstein-Podolsky-Rosen paradox. The perception of radiant energy as light is the basis of the observations in these paradoxes. For this reason, we can apply the calculus of distinctions first to the origin, propagation and perception of light. This will establish the basis for resolving these paradoxes and the conflicts between relativity and quantum mechanics.

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