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

In our application of the calculus of distinctions to symmetric contractions in the real world, we have already introduced opposites in the interpretation of distinctions. We have recognized that, due to the perfect symmetry of the void and the natural tendency of all perturbed substance to return to equilibrium, a distinction of contraction implies the existence of a corresponding opposite distinction of expansion. The following procedure is presented to help the reader achieve more facility in dealing with calculus of distinctions expressions.

If we designate the distinction of contraction as positive, then the distinction of expansion becomes negative. Thus the existence of both distinctions in the same substantial region yields an overall value of , the void. This may be expressed symbolically by the equation:

where the plus and minus signs indicate which distinction is contraction and which is expansion. This suggests a way to evaluate any expression and check the validity of equations. If we think of expressions as regions partitioned and divided by distinctions, we may assign plus (+) to distinctions in every odd depth and minus (-) to distinctions in every even depth. If the pluses and minuses all cancel out, the value of the expression is . If they do not, the value is .

The number of distinctions must be either even or odd. If the number of distinctions in an expression, i.e., the depth of the region, is even, the value is .

EXAMPLES:

, , etc.

If the number of distinctions is odd, since the first distinction has been defined as a contraction, the value is .

EXAMPLES:

, , etc.

Consider .

Both distinctions on the left-hand side must be marked as plus (+), since they occupy the same depth of the region they partition. Thus:

therefore may be marked

Similarly,

and becomes .

With these simple rules, we may now evaluate any finite expression. For example, the expression shown earlier to have the value :

Canceling out all the " - + " pairs that we can, we have a "+" left over. Therefore, the value of the expression is .

Consider the equation:

Assigning pluses and minuses, we have:

Canceling one pair on each side, both sides have the value and the equation is valid.

This validation procedure may be extended to the algebra of distinctions. We see, for example, that the value of the expression depends upon the value of the variable X. Since indicates that the two distinctions have no effect on the value of the expression. In fact, we have demonstrated that X.

We have seen that the calculus of distinctions has the capacity to include representation of positive and negative values. What about other types of values?

Consider the equation: X

In the calculus of distinctions, as defined so far, this equation is contradictory or indeterminate. Substituting values for X in the right-hand side of the equation, we get:

If X , (either positive or negative)

then X , a contradiction.

If X ,

then X , a contradiction.

However, the equation X , is not meaningless. In fact, it is no more meaningless than x2 = -1 in numerical analysis. It may be shown that it is analogous to this equation.

Since the calculus of distinctions is non-numerical, we will avoid calling the solution to X imaginary, even though it is analogous to x2 = -1 which produces imaginary numbers in numerical analysis. If we give and the designations true and false, respectively, we notice that, assuming X to represent a true statement, results in the conclusion that it is false, and vice-versa. We have a symbolic representation of the classical liar's paradox. However, we must recognize that the statement is paradoxical only in respect to the rules that we have established. We first assumed two states of reality: a state distinguished by the differentiation of one part from the rest, and the undistinguished or void state. This assumption allows only three types of statements: true (distinct), false (void) and meaningless (indeterminate).

The experience with numerical analysis reveals that imaginary numbers are not really imaginary since they prove to be useful in the solution of real problems. In the infinitely continuous universe that we are in the process of defining with the calculus of distinctions, the same sort of situation proves to be the case. In Laws of Form, G. Spencer Brown declares that we must allow not three types of logical statements, but four: true, false, imaginary and meaningless.2.

Application of the calculus of distinctions to the universe which we have assumed to be a manifestation of an infinitely continuous reality reveals the need to redefine these types as true, false, transcendental, and meaningless. The solution to the equation x2 = -1 was called imaginary because it could not be visualized in three-dimensional space. Since three dimensions were traditionally considered as the obvious extent of the real world, the value of was designated as imaginary. We now know that reality is more complex than this and it requires more than three dimensions to describe the real world. Therefore, it is more appropriate to designate the solution of X as transcendental. That is, transcending three dimensions.

THE DISTINCTION OF TIME

If the variable X represents spatial extent, then the variable y must be used to represent extent in some other set of dimensions, since it is undefined in the three dimensions of spatial extension. This other set of dimensions must relate to the substance of reality, since it arises in the algebra of distinctions which we have based upon a distinction of contraction in the substance of reality. In our experience of reality there is one measurable parameter other than space that affects the perceived forms of substance. That parameter is time. We know that in the real world the forms and the distribution of substance, i.e. the patterns of distinction, change with time.

We know that our awareness of reality has limitations. The physical organs are finite and, therefore, only capable of receiving and transmitting finite ranges of information in various forms of energy, notably light and sound waves. Our organs of perception select only those forms of energy which they are capable of detecting.

For the purposes of discussion and understanding, we separate certain aspects of our perceptions from the whole. The strength of the calculus of distinctions lies in the fact that we may apply it as easily to three dimensions as to one or two. While two-dimensional calculations may be difficult in conventional mathematics, especially in three or more variables, calculations in three dimensions or more are intractable except in a limited number of special circumstances. But with the calculus of distinctions, we may begin with three-dimensional distinctions. Conception of the primary distinction and all subsequent distinctions as three-dimensional does not encumber the calculus of distinctions, but rather empowers it with the capability of describing the world we perceive without dismantling it.

Thus, if X is taken to represent a three-dimensional distinction, there is no reason to assume y must represent any less. Albert Einstein and G. Spencer Brown both assumed the time variable to be one-dimensional. This is because they were conditioned by training to think in terms of one-dimensional distinctions. Also, the type of consciousness we experience selects a single one-directional timeline in order to make sense of events in the three-dimensional arena of perception. This does not mean that others do not exist. We shall see, as we proceed to develop the calculus of distinctions, and the comprehensive understanding of the nature of reality that its application engenders, that there is considerable evidence that time is also three-dimensional.

By use of the basic expressions and Theorems 1 through 8, we developed Ex. 9:

A

If we let a = X, a three-dimensional distinction in space, and b = T, a three-dimensional distinction in time, then Ex. 9 becomes:

Ex. 11

where X = f(x,y,z) and T = f(t1,t2,t3) .

But, Ex. 11 is derived from

Therefore, A is self-referential and a joint distinction in time and space. Since we have identified the extra-spatial, three-dimensional field as the temporal field, we set

T = and Ex. 11 becomes:

Ex. 12.

Thus A now represents a complete six-dimensional distinction in the infinitely continuous expanse of reality. In Ex. 12, we have developed a calculus of distinctions expression that describes the primary distinction, A, as a self-referential distinction in the substance of reality defined in three spatial dimensions and three temporal dimensions. This substance manifests itself in the universe as consciousness, matter, and energy. The interaction of these three forms of substance creates the relative measures of time and space.

We have now demonstrated that the calculus of distinctions has the facility to represent all the aspects of reality. More than that, it relates an act of consciousness, the making of a distinction, to the creation of the perception of reality: the world and the universe. This makes it a very powerful tool for understanding the nature of reality.

Since the act of distinction requires the existence of consciousness, consciousness must be innate in the void. The simplest way to visualize the creation of a universe such as the one we perceive is to propose that consciousness is itself the substance of the void. It is then unnecessary to introduce additional sources or substances. In the perfectly symmetric, infinite expanse of the void, consciousness may act upon itself, creating distinctions that bring everything into existence. In Laws of Form, G. Spencer Brown speculates that an entire universe is created when a region of space is separated from the rest.

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