The 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.

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

- The
Experiential
case is relatively uninteresting because the kind reader must be aware of their own existence or else they could not be perusing this page.
- The Representational case is extremely interesting because it is the unique instance where the External and Internal Forms of an Object are identical.

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

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**:

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

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:

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 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:

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

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

1;

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

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

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

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

These *Initial* forms of the Mark allow Number and Order.

Any visual form which

indicates a distinction between, say,

an internal and an external space,

has a strong
of

which preserves the
of the Mark of Distinction.

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
of
which preserves the
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
of the *Initials*.

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.

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

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

The kind reader is invited to test evaluate a few of these **Arithmetic Expressions** of a Non-Numerical
.
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**.

Given the formal equivalency and between

The kind reader is requested to consider the following pair of
**Arithmetic Expressions**:

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
among all
Arithmetics
can be used to solve the second equation.
Since it comes from the **Calculus of Indications**, the strong
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
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
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
of Distinction** which results from the
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 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 , such that:

- There are two
**Real Operands**which can be represented and experienced; - There are two
**Imaginary Operands**which can not be represented, but which may be experienced on a meta-, or complex, level; - Only the complete experience of a Distinction can preserve the of the Distinction.

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

- Statements such as
*This statement is false*; - The technique of Zen Buddhist
*Koans*; - The
of
*The Recursive Conditional Arithmetic Expression*; - The
of
*Logical Assertion Processing*in Computer Languages; - The Relationship between
*Application Programs and Operating Systems*; - The Formation of the Formal Form for the experience of Objects;
- The Spectrum of Possible Forms of Interaction between Two Objects.

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

hits since 95/11/28. Updated 96/10/13.