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Copyright © 2006-2009 Tony Giovia

23. Multiple Level GO Mechanics v2.5

23.1 - A First Level contextual relationship is formed between two Geometric Architectures when one or more dimensions in the construction of one GA exist identically in the construction of another GA.

a)  A First Level relationship is formed when contexts organized by different rules form relationships by directly sharing one or more dimensions via valence.

23.2 – A rule is a unique logical or mathematical structure of dimensions.

a)  Every rule is a unique structure defined by the logical or mathematical relationships of its composing dimensions.
b)  The physical structure of a dimension is defined at least by its Geometric Architecture.
c)  Every rule is composed of Geometric Architectures.

23.3 - A filter associates or disassociates dimensions based on the logical or mathematical laws organizing the filter. By so doing, filters associate or disassociate contexts, and associate or disassociate Ideas.

a)  A Dominant Focal Geometric Architecture (DFGA) is the GA within any context to which other GAs in the context refer. The DFGA is the core physical structure to which other GAs in the context attach. (Definition)

b)  A Point of View (POV) is one or more Base Geometric Architectures logically connected and used as the primary filter for other ideas. Like the subject of a sentence, or the foundation object in an image, the focal point of the Point Of View is its DFGA.

23.4 - A Second Level contextual relationship is created between two Base Geometric Architectures when at least one dimension of each BGA is related by a rule independent of the rules organizing the first two Base Geometric Architectures; that is, by a single mathematical or logical rule separate and different from the logical and mathematical rules organizing the first two Base Geometric Architectures. The Second Level related dimensions are never identical to the shared dimension or dimensions creating the First Level contextual relationship.

a)  Every rule defines a context.

b)  Rules can be combined to form complex contexts.

c)  Complex contexts consist of a Dominant Rule and one or more Recessive Rules.

d)  The logical and mathematical operations of the Dominant Rule are the primary filter in any complex context.

e)  The logical and mathematical operations of Recessive Rules in a complex context are filtered through the logical and mathematical operations of the Dominant Rule.

f)  Dominant Rules and Recessive Rules are composed of dimensions.

g)  In a Second Level contextual relationship at least two dimensions are related by a rule different from the DR and RR in a complex context. This new rule becomes an additional RR in the complex context. Those dimensions are never directly shared by valence between the DR and the original RR.

23.5- A Third Level contextual relationship is created among the Dominant Rule and two Recessive Rules of a complex context when dimensions within Recessive Rules (and not within the Dominant Rule) form one or more First Level relationships. In these cases the DR indirectly shares dimensions with both the RRs via its directly shared dimension(s) with one or both of the RRs.

23.6 - Individual GAs in a container object must share one identified dimension with all the other GAs in the container, but they may also share additional dimensions with one or more other GAs in the container.

a) The Dominant and Recessive Rules of numerical container objects, also know as sets, are always the product of an Objective Point Of View. (Definition)

b) The Dominant and Recessive Rules of non-numerical container objects, also know as sets, can be the product of either an Objective Point Of View or a Personal Point Of View. (Definition)

23.7 – Logic and Mathematics associate dimensions by assembling physical dimensions into physical form factors.

23.8 – Logic and Mathematics are the basis of both “hard” sciences like Physics and Chemistry, and also of “soft” sciences like Sociology and Psychology.

a)  The dimensions of “soft” sciences are physical dimensions.

b)  Physical dimensions obey physical laws.

c)  The dimensions of soft sciences must obey physical laws, just as do the dimensions of hard sciences.

 

First Level relationships involve the direct sharing of dimensions in the actual physical construction of the participating contexts/GAs. Contexts participating in First Level relationships often consist of many dimensions, and except when contexts are identities many of those dimensions are necessarily not directly shared among First Level relationship partners.

A First Level relationship between two contexts may be said to have two “angles” – two Points Of View – with one POV Dominant and the other Recessive. The POVs are different because while each POV shares one or more dimensions with the other POV, each POV additionally contains dimensions filtered out by the other POV. 

It is the relationships among the shared and not shared dimensions that create Second, Third, and higher Level relationships among contexts. These are the relationships we speak of when we discuss the multiple “sides” of a question, or the “angles” of a situation, or the “weighing” of a decision.

To illustrate a First Level relationship, consider the dimensions in the following two contexts: “A long, wide airplane.” and “A small, light airplane.”

Table 1

Assume both contexts filter perceivable dimensions by associating or disassociating dimensions as they logically apply to the dimension “Airplane” – in other words, “Airplane” is the Dominant Focal Geometric Architecture of each context. All logically associated dimensions must refer to the DFGA, and all associated dimensions plus the DFGA create the Point Of View. If the DFGA of Context 1 was “Long”, then “Long” would be the GA to which all dimensions logically refer – and “Airplane” would be a logically associated dimension drawn from the perceivable pool of dimensions.

There are three points to consider here. One, all dimensions can be viewed as physical objects. Two, the GA “Airplane” – the Idea “Airplane” - has the identical physical form in both contexts, separate and apart from the unique physical GA forms of Long, Wide, Small and Light. Three, the actual physical form of the GA “Airplane” varies from human mind to human mind – the GA of “Airplane” conceived by a child who has seen an airplane on television is very different from the GA of a pilot. All GAs are Personal Point Of View dependent. However, the basic form of the GA in any particular mind remains unchanged, because a definition is exactly its GA, and vice versa. Changing any Idea’s definition even minutely changes the Idea’s GA. An atom can be described in general terms so that it is clearly identified as an atom, but multiple snapshots will show that identical atom with electrons in different positions, the changed relationship of neutrons and protons as each spins on its axis and in space, and so on. So too with GAs – they are dependent on the perception of the particular human mind at the time of conception – the snapshot – the “first impression” - while still retaining a unique Platonic form that grossly differentiates an airplane from, say, an automobile.

In Table 1, both contexts share one uniquely defined dimension – Airplane. Relationship “A” in Table 1 above identifies a First Level Relationship between the two contexts based on their shared dimension. Now some may concede the unique physical form of the GA “Airplane” in a particular mind, but may question that the two contexts above are joined at that GA. They envision multiple instances of the unique GA that defines “Airplane”, and multiple instances of the GAs Long, Wide, Small and Light – all existing simultaneously in the same mind. Consequently, they do not concede that two or more contexts can or must share the same dimension – instead, they envision multiple copies of the same unique GA in the same mind. Or, perhaps, they envision a “Master GA” for each unique Idea that issues copies of itself when perception finds a match.

These are fair questions. Let me answer them in turn. There are those who might argue that there are multiple copies of unique GAs in the same particular mind, and therefore it is not required that two contexts share dimensions in their physical construction because there are multiple copies of each dimension to be had. Essentially, each context exists in its own level, separate and apart from other contexts, and any communication among contexts is handled by a logical and mathematical process layer – a protocol, if you will. While I can’t and don’t deny that this scenario is possible (even probable), the introduction of a communicating layer between dimensions actually does not seriously affect or change the shared dimension theory. In a physical world, the communicating layer (itself a GA) becomes part of the structure of both contexts. The key point here is that communication is a physical exchange of energy, requiring a physical pathway between dimensions and their associated contexts. Arguing whether a dimension is directly shared among contexts, or whether there is a fixed GA connecting the contexts at the point of the identical dimension, is perfectly legitimate, but it has no practical effect on the direction of this presentation.

The “Master GA” theory is a variation of the same theme. Issuing “working copies” of a GA from a Master List (as opposed to a single GA exposed with each use) is an attractive Idea, however it in no way denies that a single copy can be shared among multiple contexts (whose dimensions are themselves copies, except when the GA is new to perception and vaguely defined as a conglomerate of possible existing GAs – think of mystery and horror movies). If there is indeed a Master List, then there must also be a feedback mechanism to force a change to the Master GA – or merely an addition to the Master List – when perception changes the definition of a GA. While a feedback mechanism is probable in this scenario, how feedback might work is beyond the purpose of this presentation. Bottom line, a Master GA theory does not negate the sharing of dimensions between or among contexts.

We will therefore proceed with the theory that a single unique dimension, whether authentic or a copy, can be physically shared in the construction of two or more contexts. This has implications for the physical manifestations of Logic, because it is a look into how Logic physically contextualizes knowledge, into how Logic actually “works” in a spatial, dimensional sense. In Context 1 of Table 1, the physical GAs “Long” and “Wide” are logically associated with the GA “Airplane”. Since we are dealing with physical, dimensional objects, the implication is clear: Logic is a mechanical process that physically assembles uniquely defined Geometric Architectures into a single form factor.

In other words, whenever we say Logic (or Mathematics) associates dimensions or Ideas, in a mechanical sense we mean Logic is assembling dimensions into a larger form factor. These GA connections are always physical connections within any defined context. An airplane that is long and wide is physically different than an airplane that is small and light – in both a mind and on an airfield. Logic is a physical science that must obey the same physical laws as other “hard” sciences. And because logic is the basis of all disciplines, including all the so called “soft” sciences (psychology, sociology, art, literature, etc), all science now has a “hard” dimension.
Logic thus reveals itself as a natural mechanical process of the Universe. In my PPOV, there is no current way to know if Logic in its natural state is an aggressive force, a passive mechanical procedure, or neither a force nor a procedure – rather, it is a static unified structure. If aggressive, Logic actively searches for physical dimensions that physically fit together – its very GA structure would seek to organize all dimensions into a unified whole. If passive, logic is simply a description of physical dimensions “bumping” into each other and physically connecting if there are appropriate ports to do so. If Logic is a static unified whole, one huge GA, then it is the perceptual limitations of the human psyche that makes Logic appear to be active or passive as we assemble and disassemble the finite dimensions visible to us. The human psyche mirrors this ambivalence – “Answers Do Exist” (aggressive) versus “Que sera, sera” (passive, static).

A rule is a structure of logically or mathematically related dimensions – which can be viewed as an actual physical structure. In the case of a complex context, there are two or more rules physically connected into a geometric form factor. Each rule is a logic filter, assembling and disassembling dimensions according to the physical “ports” their logical form factor allows. Yet each GA composing the logic filter is itself part of a larger nexus of dimensions – “long” cannot be understood without a logically associated dimension of “small” – meaning “small” must be included in the physical structure of “long”, accompanied by “medium”, “short”, “gigantic”, etc. With all this intertwining, how can anyone think clearly? What is the mechanism for isolating dimensions into usable form factors?

Contexts themselves are GAs, and therefore they themselves require physical connections to a wide variety of other contexts – in fact, all contexts and dimensions that exist are part of the container context that includes “existence” as a physical dimension. The World Wide Web may correctly be called the World Wide Web of Contexts, or GAs, or Dimensions. Contexts are limited only by our perception – there are a limited number of levels we humans can perceive before the assembling of dimensions becomes overwhelming and the logic machine stops. An apt analogy would be chess players – many chess players can see one move ahead, others two moves, others three or more. But at some point the complexity of moves – the complexity of dimensions – is larger than our logical perception can calculate, and the last form factor – the last logical structure – the last GA - becomes our “understanding” of a context.

Put simply, perception is a PPOV filter that displays dimensions. Logic is the mechanism that assembles dimensions into stable, physical form factors. Using perception, research scientists and artists reveal (“discover”) dimensions not previously perceivable. Using Logic, scientists and artists assemble the revealed dimensions into stable form factors that connect to existing OPOVs and PPOVs.

Logic has traditionally labeled a dimension such as an object’s color and size as an “attribute”, defined as “an abstraction of a characteristic of an entity or substance” (Wikipedia) or “a quality or characteristic of the person, thing, group, etc., indicated” (Random House Dictionary). These definitions are completely supported by physical dimensions. Abstractions and qualities are now physical dimensions – color and size are actual constructions in the GA of an object. If an object is naturally red, then the physical structure of the object absorbs all the colors of the spectrum except red – the dimensions composing the structure, themselves composed of energy, affect the energy composing light. If the object is painted red, then the paint serves the same function. An object painted red is certainly different from an object that is naturally red, and the differences are part of each object’s GA structure. My point is that if everything in the Universe can be viewed as matter and energy, then their interactions can be viewed as the physical interactions of GAs and energy.

Many complex contexts result from comparing and contrasting the dimensions composing two or more rules. Parsing a complex context results in a set of dimensions with no clear explanation of the intended relationships among the dimensions - for example, long, wide, small, light and airplane is a list of the dimensions composing the contexts in Table 1, but without rules to organize the dimensions we have only a group of objects collected into a container object. To compare or contrast dimensions one set of GAs must physically match or mismatch the GAs of the other set of dimensions. Physical filtering is just what it sounds like – like GAs match, unlike GAs are rejected (in the simplest form of filtering) .The result of this filtering is a third physical context composed of the similarities (comparison) or the differences (contrast) between the original two contexts.

Filtering operations are highly dependent on the filter. If Context A filtering Context B results in Context C, then reversing the process by having Context B filtering Context A will result in Context D. We need to clearly identify which context in a complex context is doing the filtering. This is done by assigning one context Dominant Rule status, and the second context Recessive Rule status. For the purpose of discussion, let’s assign Context 1 of Table 1 as the DR, and Context 2 as the RR. The dimensions of the DR become the Point Of View for both contexts via their logical association with the Foundation Geometric Architecture (Airplane) of the DR, allowing us to distinguish between the two separate instances of Airplane. If the POV is Context 1, then its dimensions – Long and Wide – are also the Points of View for any dimensions that Long and Wide may be logically related to.

Note that there are other relationships in the perceivable pool of dimensions. Long and Small are related to the POV “Size”, and in fact the GA for “Long” must necessarily include the GA for “Small” in its construction (and vice versa), because Long is not uniquely definable without including Short and the related Small in its definition. This POV forms another RR separate entirely from the DR, yet still related to the DR by means of the DR’s logically filtered connection to “Long”. This is an example of a Second Level relationship between the POV of Context 1 and the POV “Size”, where non-shared dimensions in a complex context form RR relationships independent of other DR and RR relationships.

Table 2

 

Table 3

 

Tables 2 and 3 illustrate variations on the preceding discussion of DR and RR. In Table 2, we substitute two types of airplane for the generic “Airplane”. Yet Boeing 747 and Piper are not uniquely definable without including “airplane” in their definition – in their GA. Therefore Boeing 747 and Piper still create a First Level relationship between the two contexts.

In Table 3, we see three relationships. A First Level relationship between “Airplane” and “Airplane” (a shared dimension); a Second Level relationship between “Flying” and “Taxi-ing” (both are part of the context “airplane” or “airplane motion” or “modes of transportation”, etc.); and another Second Level relationship between “Red” and “Green” (context “colors”). The key to determining if two dimensions exist in a Level relationship is in their definitions – in their physical GAs. If the POV used requires a particular GA be included in the definition of a dimension, and that GA is included in the definition of another dimension, there is a match and a Level relationship.

Container context rules most commonly fall into fall into two categories – mathematical sets and non-numeric object lists. Identifying the Dominant and Recessive Rules composing mathematical sets is usually easier than identifying the DR and RR of object lists because mathematical sets generally include fewer dimensions. The DR of a mathematical set might be “Name all the whole numbers from one to one hundred, and a RR for that set may be “Name all members of the set that end in zero”. The relationships are clear and unambiguous, and the DR is always an Objective Point Of View.

Non-numerical object lists can contain many more dimensions requiring many unique definitions. Let “List all the objects in the kitchen.” be the DR of a non-numerical object list. A  RR of this context – a subset - might be “List all the objects on the kitchen table”, and while most people would list obvious objects like knives, forks and dinner plates, a biologist might include assorted bacteria common to a kitchen table, a carpenter might include inset tiling, and a scientist might include sunlight. All would be correct, illustrating the multiple dimensions of a non-numerical container object. Similar issues arise with RRs subsets like “List all the green objects.” – inviting interpretations of the shades of green – or “Name all the sharp objects.” – inviting interpretations of what is sharp and what is blunt. While the product of both the DR and RRs of non-numeric container objects can be Objective Points Of View, most often they are Personal Points Of View.

Objective Points Of View and Personal Points Of View often produce different sets of relationships within the same pool of dimensions. From an OPOV, consider the following. A common example of a First Level relationship is the relationship between a mother and child. Assuming the mother has a sister, then the child has a Second Level relationship with mom’s sister (the aunt). The child would have a Third Level relationship with the aunt’s husband (the uncle). The children of the aunt and uncle (the cousins) would constitute a Fourth Level relationship. Each chain in the link from mother to aunt to uncle to cousins is one logical process from the preceding link. The OPOV is constant, based on standardized definitions, but many families produce complex relationships that blur the OPOV. Such relationships are often described in terms of a PPOV.

 

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