uiu05charting.htm "Charting a Path through the Maze"

Contemporary scientists label a generalized holistic perception of the Universe the "Unified Field Theory" and "Grand Unified Theory" and "Theory of Everything". As much an intuitive belief that such a holistic coherence even exists, drawn as it is from deduced conclusions, (distinct from any factual reality, that those beliefs hope to eventually elaborate), a Unified Field Theory, which is currently limited in its specific "scientific" application - seen as an attempt to mathematically organize the 4 observed "fundamental" Forces - is, in fact, much broader, far reaching and more significant in its ultimate reading.

We know about the atomic structuring which we are made of. And our appreciation of the incredible diversity, intricacy, magnificent subtlety and scope of creation is literally awe inspiring - on levels from the sub-atomic to the massively cosmic. Fortunately, we are of sufficient intellect to realize that all this co-existence represents and reflects an important and fundamental mutuality - between and among all things.

We examine, we measure, we explore, we think and re-think...constantly questioning and re-organizing what we learn. And even after that, we review, constantly review, to test that what we think we have learned and know, is really correct. And we stay mentally open to readjustments and redefinitions, as we incorporate more information, variables, etc in a coherent conceptual picture. A process which in and of itself is a representative analog of the general dynamics working at large. A Gestalt - a world view , a coordinating synthesis of separate things forces perceptions and ideas - which self expresses and self defines.

In one sense, making conceptual connections does indeed expand the world for us, enabling us to do and be and know "more" than what existed before those connections were made. Sometimes, however, when certain connections are established - others may be precluded.

This is not an attempt to put a value judgement on that fact. Sometimes resultant affects are beneficial, sometimes neutral, sometimes negative, sometimes absent. Sometimes other benefits are delayed or resequenced. In this way, the "opposite" of a "positive" situation cannot be assumed to be only "negative".

-A ("not A"), can be a whole spectrum of "other-ness", that includes "neutrality" and myriad other contextual possibilities. When something "A", does "not exist", this status of "being" is a temporal and contextual relationship. It merely indicates "not present at this time", but it does not specify or qualify the conditions that would allow for the existence of an "A"....the multiple co-conditionals of "how" "where" and "when" (apropos for any give subject), let alone the panorama of possibilities of "not A".

Importantly, this is our first clue as to some of the limitations borne in Set Theories which prioritize the simplistic situation of a spatially based concept of "present/not-present" (the Aristotelian mode), rather than recognizing that a better "Set" Theory or "Logic Theory" is more properly a cognition and affirmation of relationships that are essentially Temporal (time oriented), with many-faceted conditions contributing to each result in a procedure. This is an extremely difficult jump for most of us to make. The bulk of our complex information age technology and social organization has been developed on the foundation of switching theory that relies on diverse combinations of simple on/off switch gates - binomial processes - elucidated by Alan Turing (the Turing "Machine";1936) and developed into usable form by the Information Theory of Claude Shannon in the 1940's (Mathematical Theory of Communication; Bell System Technological Journal; July, October 1948). According to 2000 years of Aristotelian thought, either something "is" or it "isn't". Plain, simple and "obvious". Except. There might be reasonable alternatives. Like the ones mentioned above which are dependent upon extraneous conditions.

In the last few years a "new" technique has appeared. It was dubbed "Fuzzy Logic" by it's originator because it relies on statistics and indistinct boundaries rather than on-off or either-or conditions. "Probability" gives weighted importance to diverse yet correlated factors across an infinite spectrum of values and therefore gives a greater flexibility in output results. It principally grew out of investigations into Artificial Intelligence in the 1960's, enunciated by Lotfi Zadeh <L.Aliaskerzadeh> (Berkeley, 1965; "Fuzzy Sets",Information and Control, pp 338-53) who adapted ideas developed by J.Lukasiewicz (circa 1920's)....and then became delineated and applied to real world technology in the recent 1980's. (Primarily in Japan, it should be noted.) Fuzzy Logic marks a strong departure from the logic methodology used by humanity since Aristotle. It's appearance and validity show us that the tried and true - no matter how long in affect - may not necessarily be "the best" way or "the only correct" way.

If we look more clinically and critically, though, at the "dynamic functioning" of Fuzzy Logic, we note that the scientific community has been using it - more accurately, a specialized form of it - for generations, not just the current crop of feedback-control applicators and Artificial Intelligence theorists.

A mathematical logic form named in honor of its inventor George Boole (ca.1854) called Boolean Algebra, was one of the earlier efforts to blend the "absolute" with the "statistically probable". In my opinion, George Boole and Huntington (1905) were the first propagators of "Fuzzy Logic" even if their focus was more narrowly defined and parameters constrained. Boolean Algebra was in effect the very first exposition of Stochastic Logic. It predates and defines that statistical portion of the methodology claimed by Fuzzy Logic. Obviously, Fuzzy Logic embellishes and improves immensely upon their initial work. So, Stochastic Logic can be thought of as a title/name that better includes Fuzzy Logic and Boolean Algebra, as I shall explain here.

1. a (dot) bar-a (equals) 0 a • ä = 0

2. a (plus) bar-a (equals) 1 a + ä = 1

Traditionally, Boolean Logic has focussed on these operations as producing 2 and only 2 binary-states ... 1 or 0. This was reinforced by the Aristotelian evaluation of function (2.)...there is equal "probability" of "a" or "bar-a"{not "a"} ... something existing or not existing ... a homogeneous binary "or" switch. Also, Boolean Algebra differs from standard algebra by not being "commutative". That is, under standard algebra a+b = b+a and ab = ba. In Boolean Algebra, the sequence makes a distinct difference in "information" ...there is a variance - something uniquely different to distinguish between two "similar" statements. a+b … b+a and ab … ba.

Inherent in Boolean Algebra though is a wider latitude of definition of "stochastic probability" and therefore competent usage which Zadeh defines and makes wide practical use of with Fuzzy Logic. The probability of "a" or "not/bar a" can vary. Like Hamiltonian functions.

1. The probability that a (and) not-a can co-exist simultaneously, is zero.

2. The probability of a existing, (plus), the probability of not-a existing adds to one.

Under Boolean Logic the probabilities of statement 2 are equal. Under Fuzzy Logic they need not be, nor do the number of parameters have to be limited to two. As long as the probabilities sum or exclude to an appropriate value, where the number of components are not binarily restricted nor are constrained to result in "zero" or "one". It is all one logic system with Boolean being a subset of general Zadeh Logic. It is STOCHASTIC LOGIC.

When we look at those fundamental a-priori relational expressions...for analog qualities we can apply when examining "existence"...we immediately see that they are variant aspects that have already been used and applied in Euler relationships of complex numbers and in the logarithmic transcription of functions (exponents, trigonometry, etc.).

In our search for "boundary functions" which would clarify Aristotelian logic carried to its extreme by Gödel, Boolean algebra accomplishes this very exactly. It establishes mathematical "environments" and their relations. It redefines "identity functions", both from "within" and from "without", and thereby bridges across the limit-wall of Aristotelian-Gödelian Formalism. It also takes us one step closer to coordinating quantum mechanics with relativity.

By reclarifying the nature of the operators used in these equations, we can state: when a domain is expressed in the exponent portion of a mathematical statement, it can have a permitted value state of "zero", where that "zero" can also be represented by substituting "the derivative of a constant", and, we establish not only a domain (from -4 to 4) in the "base" location, but co-establish domains (from -4 to 4) exponentially lower and higher in relation to the base being examined. These are Nested Cantorian Infinities, and the boundary conditions are the two Boolean expressions of stochastic existence (using the expanded meanings applied according to Fuzzy Logic).

The singular expression which blends Boolean Algebra with normative non-Boolean Algebra looks like this:

"c" is an adjectival modifier of the Inertial Reference Frame "J". "z" is the collateral (exponential) Reference Frame. When "z" takes on a value of zero, its presence is maintained by being the derivative of a constant present in the collateral Reference Frame.

The "dot" function is alternately expressible as the "plus" function (!). When "z" happens to be "zero", we can recall and insert by substitution an alternative form for describing "zero": 0 = DW/dx. Zero is the derivative of a constant. What this accomplishes is to expand the presence of derivatives into adjacent exponential and sub-exponential domains. Our first glimpse of multiple Cantorian Infinities, and Exponential Calculus.

A function which is/equals "zero" ("z" in the first instance) can be substituted for a function which is/equals "one" when "J" is "one". A valid mathematical sentence when "one" and "zero" are reasonably "interchangeable" ... sic: "equal".(!)

Between the exponent levels, the relationships are Boolean. Within domains, the functions are algebraic-normal. This would explain and allow for entropic behaviors as being negentropic in corresponding exponential domains (alternating in +/- signs "relative to each other"), yet each domain is always self-referentially "positive". In this new way, + or - is a "comparative state" for operational gradient directions, instead of the traditional usage of a value-state within a continuum. Within a continuum, emphasis is on Cartesian "positive" or "negative". Across continuums, emphasis is on the absolute quantum values, and +/- indicate the direction of the entropy gradients.

"Fuzzy Logic" is not new. It is an old friend. And we must applaud Zadeh's intellect for making us recognize that.

Before proceeding we should make one special note about Boole and his logic system. Boole was a scholar and teacher whose ideas were deeply rooted in Aristotelian tradition. He stated very clearly in his 1854 book that it was his wish to formalize Aristotle's ideas into mathematical notation. The wry twist of reality is that his formidable efforts which gave us the foundation for our information age technology also opened the door for Zadeh and the larger panorama and improved techniques of non- Aristotelian Fuzzy Logic. Boole (via Huntington) brought a clarity to Aristotelian thought which has enabled us to surpass it.

Other important considerations are present here. They involve Information, entropy, noise, Shannon, Boole and commutivity. Shannon's assessment of "noise" equating with entropy (defined under calculus summation equations) means we have to explore commutivity in that regard. Imagine that we could only see one side of an equation statement, "ab" for example. We don't know what is on the other side of "equality". There could be another ab; there could be ba. That lack of "information" is considered by Shannon to be "noise", scattered knowledge, entropy. Yet, under Boole (standing side by side with Einstein who said any frame of reference, any point of view, is as valid as any other) we could just as easily be observing "ba" instead, and call the unknown "ab" as the "noise". Both are information in and of themselves. What this shows is that there is another factor of importance: sequence or 'ordering'. In commutative relations, the sequence is secondary to the content. In non-commutative relations, sequence is an equally important distinction. And sequence is a phenomenon knowable only through the medium of Time - i.e., temporal displacement. Temporal continuums can therefore be a source of information, separate from other coordinate factors. It is an environment that enables discrimination and discernment. It embodies its own domain of information.

Also, the form of any expression-statements reveal or codify another independent order of very explicit "information". The language form carries both indisputable content and indeterminate (yet still valid) content. Sometimes the data is explicitly present. Sometimes an inherent variable (information distinction) is masked. Not because it isn't there, but because it is not enunciated in every expression. Implications are present based on accepted or presumed mutual conventions.

Benjamin Whorf's concepts highlighted in the Introduction become exceedingly clear here. Shannon examined phenomena and factors of information transmission systems, attributed them with meaningful characteristics, and labelled them with corresponding relational mathematical values ... and gradients of differing direction. In other words, even hallowed "mathematics" is properly viewable as a malleable "language" ... open to and subject to interpreting and re-evaluating specific meanings ... those "intended" compared to those "predominating", and even compared to those "present but unrecognized before". Experience creating "expression". And "expression", focussing and re-enforcing those earlier experiences, until the experience/expression matrix is re-evaluated, enabling a more recent and expanded "experience" ... redefining the "expression".

The Boolean relevant format is, in another sense, an alternative rephrasing of the idea that Reimann-Einsteinian "reference frames" will dictate their own very specific and unique observations and function values. The analog here was not readily apparent before as being equatable with Einstein's evaluation, since his work concentrates on the relativity that becomes apparent when different values are generated around a fixed constant and a singular uniform relationship expression. The extension of Whorf's "relativity" that I am applying here deals instead with the "relativeness" of how the information present is describable in several different relationship/expressions. Some technique of translation is available that allows things to be equal if not identical. The essence of both is to say that a change of any aspect of an event system will produce a change in the quality or nature of the information transmitted/transcribed, and therefore observed. A symmetry type ffunction.

For example, Einstein's famous thought experiment about a train with one observer riding and one observer standing next to the tracks can now be re-evaluated to correlate with the premise here. The one riding on the train stands in the last doorway of the last car. As he reaches the location near where the track side observer is standing, he throws a ball horizontally straight back along the straight line path where the train came. Whether by luck or good effort, the ball moves so that its horizontal motion becomes uniform in front of the track side observer and the ball moves only in the vertical direction exactly in front of the ground site observer and bounces until it come to rest. The train rider sees the ball "accelerate" away from him until it reaches a constant velocity receding into the distance. The track side observer sees the ball "decelerate" until it comes to rest (constant velocity = zero). A single activity embodying two "opposite" transition motion values. The difference being the "reference frame", the "point of view", the reference domain. Exponential expression/relationships are similarly distinct and "relative" in their motion gradient values. Where as, the functions are applicable everywhere.

To rephrase: The "Objectivity" of interaction/experience is in the universality of the dynamics. The "Subjectivity" of interaction/experience is in, or due to, the specificity of the participants.

"What" we see and say is fortified by previous experiences and the extent or restrictions imposed by environment. The old psychological saw about a glass half filled with water is an apt example. An "optimist" sees the glass half full. A "pessimist" sees the glass half empty. A nice Aristotelian "either/or" (A or not-A). A nice and easy classification technique.

Well, I see things quite differently. From my experiential frame-of-reference I see a glass completely full!!! The glass is filled with things I need to survive ... water and air. If either of them are taken away from me ... I die. Not only that, but the glass is an active participant in the scenario and plays the role of a limited localized "boundary" which gives me a lot of additional information about my world. If the glass is held open end to the sky, it stays half+half filled. In order to do that where the water predominates, the glass must be reversed..open end down. The thinner the walls and less mass/weight of the glass involved, the more I measure that there is some equal yet opposite-acting condition that tends toward the surface interface of the air-water volumes as some kind of meaningful "boundary". An "equal but opposite" force is functioning in my world, the intensity varying about a stability point at the interface. Other variables - including subtle changes or obvious ones - e.g.: density of the mediums, temperature, open glass, sealed glass, etc. - provide an immense amount of information. Including, apparently, whether I am an optimist or a pessimist, or an insatiable learner who never tires of seeing the possibilities of things!

The ultimate point of all my writings is that we must push ourselves to look at all possible environments and relationships as we try to formulate an "ultimate gestalt". This approach will eventually guide us to understanding Time as a plural-dimensional phenomena. That is, Time as a richer phenomenon than is currently appreciated or used. It has aspects and impacts that we can newly discern, if we proceed carefully and openly. Time is not some enigma. It is not a bland monolithic one-dimensional featureless continuum. It factors in intimately with every functional event and relationship in the universe in ways at once more subtle and more dynamic then conceived of in existing schemas, and its aspects of self-reflexive multiple-dimensionality cannot be dismissed, even when we focus of such things as "forces, relationships, parameters, derivatives, or ffunctions at an "instant" .... when the "no-time" temporal point-instant defines a situation when "customary non-usage" allows us to ignore the very presence of Temporal continuums. Most important, elucidation of relational situations that are co-relevant with Time, can take us farther than ever before in understanding the universe we exist in and are part of. A Universe of Information.

When scientists and theoreticians propose multiple dimensions, the higher dimensions they refer to are essentially space- relevant. It is a kind of compaction that non-mathematicians have difficulty picturing mentally. As I will describe further along, there is another way of considering multiple dimensions, and though I can't promise them to be any easier to picture, their existence as plural temporal domains, leads to extremely useful results that help resolve certain disparities in our growing coherent ideation of the Universe in toto. To spark a recognition in you, of the valid sense of these last statements, I ask you to hold the following question in your thoughts: Why, if "time" is spoken of in all human discourse as one-dimensional (the single "river" of Time), is it always descriptively depicted in mathematics as a three-dimensional time-"cone"? (!) ... a multi- dimensional geometric form. Both are valid. Something wondrously intriguing is going on here.

We must begin by looking at the mental tools and ideas we already have, to see if we can find some way of cleaning house. Find some way of clearing up these kinds of disparities and disjointed ways of thinking. Choose a path of clear and consistent definitions. Adjustively adapt notations or perceptive biases. Allow the free flow of thoughts. Let diversities exhibit their functioning in natural Accords. Before making a frontal assault on the language of mathematics in the hope of clearing and expanding our conceptual horizons by expanding the true factuality of several multiple dimensions of spacetime ffunctions, I feel compelled to give you a deeper appreciation for the power of languages in general. Once we feel comfortable about mastering our communication systems at least to the extent that they have already molded us, we can go back and clarify the language of mathematics in order to take a quantum leap toward the pending millennia of human thought.