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Formal Formulations


The Form and Substance of
Non-Numerical Arithmetics



The Substance of a Non-Numerical Arithmetic is the intellectual concept of the "Existence of Objects", whether the Objects are real or imagined. Needless to say, any such discussion of existential import ultimately has a spiritual genesis.

Laws of Form The algorithms for indicating existence may have a Form which is either Experiential or Representational.

The discussion which follows is based on the work of Mr. G. Spencer-Brown which was first published in 1969 under the title "The Laws of Form". Regarding this work, Bertrand Russell commented that:

"In this work, G. Spencer-Brown has succeeded in doing what, in mathematics, is very rare indeed. He has revealed a new Calculus, of great power and simplicity."

In the "The Laws of Form", Spencer-Brown develops the axioms, canons, conventions, and other formalisms necessary to construct what he calls the Calculus of Indications, in a completely rigorous manner. This Calculus of Indications may be immediately interpreted as a Boolean Arithmetic.

The development and application of the Calculus of Indications is divided into the following sections:

* Debunking Boolean Algebra
* The Initials of a Non-Numerical Arithmetic
* Expressions of the Arithmetic
* The Canonical Form of Distinction
* Related Topics

Finally, the kind reader can always refer back to the discussion of ===========================

Debunking Boolean Algebra

In order to fully appreciate what is going on here, The Homeless Mathematician must inform the kind reader that what they may have thought of as Boolean Algebra is actually a grand misnomer and incorrect usage of the term "Algebra".

What is commonly called Boolean Algebra is actually a collection of discrete function tables which represent the behavior of arbitrarily defined functions called "conjunction", "disjunction", "implication", "negation", etc. For example, consider three of the most common so-called Boolean Functions:

Laws of Form Laws of Form Laws of Form

Notice that the above functions are formally represented as Boolean Operators in a Boolean Expression which has a Form that Forms a strong Analogy to the Form of an Arithmetic Expression. This is a convenient use of symbolic Forms. However, it is formally incorrect to do this because such Forms import the Substance of Arithmetic Operators instead of the Substance of the Boolean Functions. The actual Form of a Boolean Operator must import the actual Substance of the Boolean Operator. The External and Internal Forms may differ, but the Substance must be preserved in order for it to be experienced.

Every philosophy and computer science student has has to memorize these tables at some time or other. Having memorized these tables, there is a classic exercise that involves proving deMorgan's Laws, namely:

not ( A or B ) = ( not A ) and ( not B )

This usually involves
an exhaustive substitution of
the possible Boolean Operands,
F for "false" and T for "true",
which the Boolean Variables,
A and B,
may assume.

For each such substitution, the above tables are then applied to evaluate each side of the equation. If both side match for each substitution, then the relationship has been proved by exhaustion.

The fact is that there was no formal calculation involved in this proof, only functional evaluation based on an explicit enumeration of values. While the Boolean Operands, F and T, may form an Arithmetic, there are no Boolean Operators. Conjunction, disjunction, and negation are Boolean Functions, but they are not formal Operators. Imagine attempting to perform Numerical Arithmetic in the same way that the Boolean Arithmetic for the above proof was done. That is to say, consider the numerical equation:

- ( A + B ) = ( - A ) + ( - B )

Proof by exhaustion is impossible. The kind reader could check this out by substituting all positive integers less than, say, one million, and then recall there are a lot of integers left to check out. Because "-" and "+" are Arithmetic Operators which have the property that:

+ A = - ( - A ),

a proof of the above Algebraic Equation ( given the underlying Arithmetic requires Operators for calculation, and the classic logic tables define Boolean Functions, not Operators. Although the deMorgan's Law is, in fact, true, it is not a Boolean Algebraic Equation. A more proper formulation for deMorgan's Laws is:

or( A, B ) = not( and( not( A ), not( B ) ) )

where not( * ), or( *, * ), and and( *, * ), designate the usual Boolean Functions.


The Initials of a Non-Numerical Arithmetic

At this stage, kind reader, it is the time, here and now to gaze on the what are called the Initials of the Arithmetic. For our common, day-to-day Numerical Arithemetic, the Initials are

0 + 0 = 0; 0 + 1 = 1.

For the Calculus of Indications, the Initials may be written, variously, as:

Laws of Form
For as much as blank represent void, this is the Unmarked State.

Laws of Form
For formal discussions, this is the perfered form to indicate the Marked State.

Laws of Form Laws of Form
These Initial forms of the Mark allow Number and Order.

Laws of Form Laws of Form
Any visual form which
indicates a distinction between, say,
an internal and an external space,
has a strong Analogy of Form
which preserves the Substance of the Mark of Distinction.

Laws of Form Laws of Form
Since curly brackets have an inside and an outside
it is possible to create great confusion
by mistaking the mathematical Set Notation of
writing "{ }" to indicate the Null Set
and, hence, writing { } = .

How silly.
Remember that a Mark is not Null.

There is a strong Analogy of Form which preserves the Substance of between the Calculus of Indications and the Boolean Operands and the "not" function. The kind reader will find this to be true, beyond any doubt, as they gaze at the actual visual Form of the Initials.


Expressions of the Arithmetic

The Initials of a Non-Numerical Arithmetic can be used as the underlying Arithmetic for a formal Algebra as detailed in the following table. Each of the 16 possible Boolean Functions of two variables are expressed as a visual combination of elements of the Initials, with Formal Parameters assuming either a Marked or Unmarked Operand. Just imagine that if something is Marked, then it must exist - otherwise it could not be Marked; if something is Unmarked, then it does not exist and, literally, nothing can be seen. The Boolean Functions are indexed from 1 thru 16, for later reference.

Laws of Form

The explicit calculation, for two of the simplest functions are as follows. Notice the strong Analogy between the Boolean Operands of "false" and "true" with the Marked and Unmarked States.

Laws of Form Laws of Form

An example that requires more steps in the explicit calculations, is give by a demonstration of evaluating the classic conjunctive function, namely:

Laws of Form

The kind reader is invited to test evaluate a few of these Arithmetic Expressions of a Non-Numerical Form . At this point, The Homeless Mathematician will note that the problem with A Non-Numerical Arithmetic that Failed is that there are no Initials with which to form an Algebra.


The Canonical Form of Distinction

Given the formal equivalency and Analogy between

  • Boolean Operands and Functions, on one hand, and,
  • Algebraic Expressions of Marked and Unmarked States, on the other hand.
  • The kind reader is requested to consider the following pair of Arithmetic Expressions:

    Laws of Form Laws of Form

    In the first case, the equation yields what are called the Imaginary Roots of the Real Number whose value is the negation of "one". Now, in physical measurements, Real Numbers are used to enumberate values of substance which we experience directly and can measure. Imaginary Roots enter into most equations which describe physical events and, in turn, have an indirect effect on the process. Consider the equations of X-ray diffraction in crystals. The precision of these equations allow us to resolve the location of indivdual atoms with in a Unit Cell of the crystal. The Homeless Mathematician wonders how many New Age crystal junkies can identify any of the 230 different space groups of synmetry which can form a crystal.

    In any case, the strong Analogy among all Arithmetics can be used to solve the second equation. Since it comes from the Calculus of Indications, the strong Analogy argues for Imaginary Roots also. In this case, the Imaginary Roots apply to a Marked State of Distinction, as indicated. Hence, the kind reader must now imagine Imaginary Marks.

    The strong Analogy between the Calculus of Indications and the Boolean Functions argues for what might be called Imaginary Boolean Operands. Thus, "T" and "F" should be called Real Boolean Operands, in distinction from Imaginary Boolean Operands which are neither "true" nor "false".

    While we can visualize the forms for the Analogy to a Marked and Unmarked State as a Real Boolean Operand, we can not visualize the form for an Imaginary Boolean Operand.

    This is the Canonical Form of Distinction which results from the Substance of Distinction. The distinction can have an Object as Marked or UnMarked, indicating existence or non-existence.

    In addition, there are two states which may or may not interact with the Form of existence or non-existence, at a meta-level. At such a meta-level, the question of existence is irrelevant until an Object is again Marked or Unmarked.

    Finally, to summarize, any Distinction has a Form , such that:


    Related Topics

    The Canonical Form of Distinction has many, many immediate and direct applications for clarifying the Form and Substance of Objects. For example:

    and many more topics which will be added in the future.

    Copyright © 1996. Formal Formulations. ~ ~ ~ Send e-Mail to comment@formal.com.
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