Formal Formulations

The Sentient Perception of Objects

Our consciousness experiences the Physical Realm of Consciousness
via
a series of organic sensory organs, devices, or faculties,
by which
external or internal stimuli are conveyed to the brain.

Specifically, the following so-called "senses" are well known.

Receptors Location Organ Common Sense Formal Activity

Special External Eyes Color Visual Seeing
Special External Ears Sound Auditory Hearing
Special External Nose Odor Olefactory Smelling
Special External Tongue Flavor Gustatory Tasting
Somatic External Skin Touch Tactile Pressing
Somatic External Skin Temperature Caloric Sensing
Somatic Internal Skin Equilibrium Kinesthetic Moving
Somatic Internal Skin Hunger Nutritional Feeding

Qualification by adjectives is the usual means for describing the of these sensory
experiences. The following lists provide a partial spectrum of what our consciousness may experience
for each type of sensory input.

Visual Seeing of Colors

black, blue, green, orange, pink, red, violet, white, yellow.

Auditory Hearing of Sounds

bellowing, blaring, bleeping, booming, brassy, braying, brazen, cacophonous, canorous,
chiming, clamorous, clangorous, crashing, creaky, deafening, dinning, discordant,
 echoing, euphonic, faint, flat, grating, harsh, high-pitched, high-toned, hollow, inaudible,
inharmonious, lilting, loud, low, lyrical, melodic, melodious, metallic, muffled, musical,
muted, noisy, pealing, penetrating, piercing, piping, plangent, purling, rattling, reedy,
resonant, ringing, rippling, scratchy, screaming, screechy, sepulchral, sharp, shouting,
shrill, singable, soft, sonorous, squawky, squeaky, stifled, strident, thunderous, tinkling,
tinny, tintinnabulary, tuneful, whooping, yelling.

Olefactory Smelling of Odors

acrid, ambrosial, ammoniacal, aromatic, fetid, foul, fragrant, frowzy, fruity, fusty, gamy,
gangrenous, graveolent, heady, heavy, hircine, malodorous, mephitic, miasmic, musty,
naris, nidorous, noisome, noxious, perfumed, pleasant, pungent, putrid, rancid, rank,
reeking, smoky, spicy, stale, stench, stinky, strong, stuffy, suffocating, sulphurous, sweet,
thuriferous.

Gustatory Tasting of Flavors

acrid, astringent, bitter, caustic, gamy, gingery, minty, mordant, nutty, peppery, piquant,
pungent, salty, sapid, savory, sharp, smoky, sour, spicy, sweet, tangy, tart, zesty.

Tactile Pressing of Touch

brush, caress, crush, delicate, gentle, graze, knead, pat, scrape, scratch, squeeze, stroke.

Caloric Sensing of Temperature

chilly, cold, cool, frigid, hot, lukewarm, nippy, scorching, sweltering, temperate, tepid,
torrid, warm.

Kinesthetic Moving of Equilibrium

dab, flick, flip, flow, fumble, hit, jab, knock, palpate, ply, poke, sweep, tap, tip.

Nutritional Feeding of Hunger

famished, hunger, ravenous, sated, starving, thirsty, voracious.

External Or Internal Stimuli and

The Homeless Mathematician notes that the use of the terms "external or internal stimuli", above, refer
to the Physical corpus of the sentient being. In all cases, the stimuli is the result of a Physical Object
interacting with a sense organ, of the Physical corpus of the sentient being.

In this respect, it may be more appropriate to use "surface", in lieu of "external", i.e., "surface or
internal stimuli", because, with respect to consciousness, everything is external.

The brain, as an internal sense organ, functions as an empathtic processor for our consciousness to
 perceive the Form and Substance of the Object which generates the stimuli. As such, consciousness
may experience each Physical Object as having both an

External , as represented in the Physical Realm of Consciousness.

Note well that in the Physical Realm , an External which represents an Object is
identical to the of the actual Physical Object. To paraphrase: "It is what it is."
Likewise, the of each actual Physical Object is unique and bestows a dignity
of distinction to each such Object.

and an

Internal , as represented by the empathtic processor for Consciousness.

From studies of hypnosis and Neuro-Linguistic Programming, the empathtic processor is

understood to have three primary modes for Internally representing the Form and
Substance of a Physical Object:
Visual
Auditory
Kinesthetic
In addition, as an empathtic processor for our consciousness, a brain may likewise
empathize with Objects contained in the non-Physical Realms of Consciousness.

Thus, the transformation of the Form and Substance of Objects between External and Internal
s governs the experiences of our consciousness. In turn, our experiences govern our interaction with
other Objects, as discussed in The Form of Interaction between Two Objects.

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Updated 96/02/01.
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 Formal Formulations

and
As The Sentient Perception of Objects
has evolved over The Realms of Consciousness,
The Paradigm of Form and Substance
serves as a touchstone reference example,
of what our consciousness experiences.

The Formal And Substantive Foundations for

The Paradigm of Form and Substance will be established by:

and
of the

and
of

and

These strong of
allows the of the Initial Distinction to be
projected across the The Realms of Consciousness
such that
it can be measured in the Physical Realm,
with ever increasing Distinction.

This triad of strong of Dialectics spans the The Realms of Consciousness, while
preserving the import of Spiritual Objects across the Realms. Thus, this trinity may serve as a
touchstone reference for all Other Dialectic Distinctions.

For a deeper understanding how Objects have been experienced, by our consciousness over the ages,
the reader is invited to explore the history of The Clarification of Form and Substance.
 Realms of Consciousness
As sentient beings,
our consciousness has the potential
to sense and experience
the Form and Substance
of an Object
on any of the four so-called
Realms, or Planes, of Consciousness.

Spiritual
Intellectual Emotional
Physical

The Paradigm of Form and Substance

An Analytical distinction,
between Form and Substance,
allows Form and Substance,
to manifest
as distinct attributes of an Object.

As sentient beings, we experience

the Substance of water
as three common Forms of water:

Solid ~~~~~~~~~~~~~~~~~~ Liquid ~~~~~~~~~~~~~~~~~~ Vapor

 In an existential or Spiritual sense,

As the is formed,
The is experienced,
In an intuitive or Emotional sense,
As the is different,
The is similar,
In a sentient or Physical sense,
As the has shape,
The has mass,
In a mathematical or Intellectual sense,
As the is correlated approximately as macroscopic energy levels,
The is measured precisely as 2 hydrogen and 1 oxygen atoms,

The above series of and span the The Realms of Consciousness and provides a
specific example of the experience of and in each of the Realms.

This distinction of and strictly conforms to The Canonical Form of Distinction,

as detailed in the treatment of Non-Numerical Arithmetic. As such, it establishes a touchstone for the
conformity, with reasoning by , of all Other Dialectic Distinctions.

In turn, this allows the formalism of and to be applied to any Object, in any
Realm. Understanding the and of an Object can provide great insight into our
possible experiences of an Object.

Formal And Substantive Foundations

In this task, The Homeless Mathematician is confronted with a "Which came first, the chicken or the
egg?" problem. Although the foundations arise from drawing a single distinction, without the
foundations, it is meaningless to draw a distinction.

Thus, for the formal construction,

of the foundation for the Paradigm,
The Homeless Mathematician, as a mathematician, must:

Firstly deal with "the question of existence" and, thusly address Experience Beyond Existence;
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 Formalize the distinction between the common Forms of Reasoning;

Formally apply the formal Forms of Reasoning to the Null Form, to allow for The Formation of
the Formal Form;

Formally extend the treatment of to allow for the experience of the and
of Objects.

Ultimately, the validity of The Formation of the Formal Form rests on its conformity to The
Canonical Form of Distinction, as detailed in the treatment of Non-Numerical Arithmetic. However,
as an Intellectual or Physical Object, no arithmetic can manifest until the Formal Form has
manifested.

The Homeless Mathematician notes that, although he was able to nullify the recursive element of the
original "chicken or egg?" problem, the recursive element persists. This is relatively free from
contradiction with, or conforms to, the of The Canonical Form of Distinction. This is a direct
and immediate validation of the Formal Form, but recursively forces a treatment of .

Fortunately, was left undefined, with a single exception: the definition of as

the of Distinction. To assure that is not otherwise specified, two conditions
are imposed:

"What is not allowed is forbidden."

This is the so-called "Convention of Intention." Essentially, it imposes an axiomatic

framework on any formulation. If something is neither explicitly allowed as an axiom, nor
derived from what is allowed, it should be considered as not allowed, or forbidden.

All experience is by .

Recall that Reasoning by Empathy is only a means of experience, and is not experience,
per se. Reasoning by always requires a definition of , and any
further attempted definition of would violate the Convention of Intention.

With Reasoning by , the of any Realm of Consciousness may be seen as

forming a weak or strong with any other .

As the and of Objects is clarified, the strength and utility of such is

revealed. The strong between

the mathematical s of the Intellectual Realm,

and

the material s of the Physical Realm,

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 is one example of such utility.

Experience Beyond Existence

The Homeless Mathematician begins with the Spiritual Realm as being:

the Realm of Consciousness

before the Beginning
of the Space and Time
of the Physical Realm
of our consciousness.

As such, direct sensory experience of the Spiritual Realm is meaningless. In order to experience itself,
or anything else, this Spiritual Realm must draw an existential distinction between "being" and
"becoming".

The Homeless Mathematician takes this act of drawing a distinction as the of Reasoning
by . As such, Reasoning by is an intrinsic Object of the Spiritual Realm.

However, it is necessary to maintain a balance among any Objects of the Spiritual Realm. Only then
will the Objects produce a net effect which is null, and which can not be directly experienced.

The balance is achieved with Reasoning by as another intrinsic Object of the Spiritual
Realm.

Formal Forms of Reasoning

To conforms to The Canonical Form of Distinction, as detailed in the treatment of Non-Numerical
Arithmetic, there are four distinct s of Reasoning as follows:

Spirituality

Empathy

Note well how and correspond to the representational s, while

Spirituality and Empathy correspond to the experiential s. Ironically, the mathematical
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 interpretation refers to the representational s as the "real roots", while the experiential s
are called the "imaginary roots".

There is an elegant subtlety in that Reasoning by Spirituality is both the source and a result of the
Initial Distinction in The Formation of the Formal Form. This serves to import a recursive element,
but also obscures the full spectrum of the Initial Distinction by an apparent collapse of four s
into three.

The Formation of the Formal Form

As a sentient being, The Homeless Mathematician constructs the formal framework for the non-
Spiritual Realms as follows:

I always imagine the Null Set

( also called The Void, or NoThing ), until,

Reasoning by , I draw a Distinction;

Reasoning by , I erase the Distinction.

This produces a Zero-Sum Game and represents a of Void without .

Reasoning by Empathy,
I empathize with this of The Void
to experience the of The Void.

This allows me to experience the and of the Void.

paradigm (pàr¹e-dìm´, -dîm´)

noun

1. An example that serves as pattern or model.

2. A list of all the inflectional forms of a word taken as an illustrative example of the conjugation
or declension to which it belongs.

[Middle English, example, from Late Latin parad ì gma, from Greek paradeigma, from
:
 paradeiknunai, to compare : para-, alongside. Also, para-1 + deiknunai, to show.]

The Formal Paradigm of essentially fixes the Physical experience of as

what can be measured and quantified. At first, this may seem very limiting and the ultimate in
Descartian arrogance. In fact, all it really does is to shift into the realm of , and there
are few limitations on . This allows us to deal with on the Physical Realm in a
precise, knowledgeable, and reliable manner. What remains aside from the is pure
and may be viewed in terms of conformity to canonical .

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 Formal Formulations
The and of
Non-Numerical Arithmetics
Exchanging Values
via "exclusive-or"s
A formal proof that three "exclusive-or" operations will
exchange the values contained in two memory cells,
using techniques from the The Calculus of Indications
Boundary Mathematics
The Calculus of Indications is now addressed by this branch of Mathematics.
Principia Cybernetica Web
This project addresses the larger issues of consciousness.
Java meets The Laws of Form
Java is another instance where the External and Internal Forms of an Object are identical.

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 Numerical Arithemetic.

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:

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:

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
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 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 Numerical Arithemetic ) is almost
trivial. An 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;
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.

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 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.

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.
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 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:
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 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.

The Canonical Form of Distinction

Given the formal equivalency and 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:

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,
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 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.

Related Topics
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.

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

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