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This is a work in progress - all rights reserved.
Copyright © 2006-2008 Tony Giovia

 

CHAPTER 16 - The Rules of Power Relationships v2.0

 

16.1 - Contexts can be constructed of one dimension, or they can be constructed of more than one dimension. (Definition)

16.2 - When a context is constructed of one dimension, it is defined by that dimension. (Definition)

16.3 - When a context is constructed of more than one dimension, all the dimensions used in its construction are related by a rule. That rule is a context defined by its dimensions. (Definition)
               
16.4 - Contexts can enter relationships with other GOs by sharing dimensions with those contexts. (Construction)

16.5 – Contexts can break existing relationships with other contexts by disassociating themselves from the dimensions shared with the other contexts. (Construction)

a) Changing one dimension of a context changes the definition of the context. (Construction)

b) In any context, associated dimensions can only be disassociated by a change in the definition of the context. (Construction)

16.6 - When a new shared dimension is added to a context, the original rule that related the original dimensions undergoes a change in definition to include the additional dimension. (Definition)

16.7- Addition is a rule that combines two or more contexts by introducing a shared dimension to the pool of available dimensions. (Definition)
16.8 - Multiplication is rule that combines two or more contexts by introducing a shared dimension multiple times to the pool of available dimensions. (Definition)

16.9 - Subtraction is a rule that disassociates a shared dimension from a context, creating a new unique definition of that context. (Definition)

16.10 - Division is a rule that disassociates multiple identical shared dimensions from a context, creating a new unique definition of that context. (Definition)

16.13- Identity Rule: For any uniquely defined context A, A = A (Definition)

16.14 – Inequality Rule: A ≠ B (Definition)

 

We are using logic as our tool for understanding and developing Dimensional Thinking. Rules and laws are equivalent concepts; however, in the case of multiple rules within a context, it is sometimes convenient to recognize a hierarchy of rules. The most common hierarchy is the Dominant Rule and associated Recessive Rules.

The operative laws of logic and mathematics are the same – there is no logical law that cannot be converted to a mathematical law, and vice versa (we show how this relates to Godel’s theorem in a later section). This means that The Geometry of Ideas, to be consistent, must completely obey all known mathematical and logical laws as they pertain to geometric objects.

Addition is a rule that joins the uniquely defined dimensions that comprise contexts; each addition creates a new uniquely defined context. To provide a sense of physicality when talking about contexts it is useful to refer to them as Geometric Outlines (GOs). Because we currently have no notion of the external shape of a GO we will, for the sake of argument, assume each unique GO defines a unique context.

There is one important difference between common mathematical notation and GO notation. Mathematically, A + 0 = A is correct notation. However, in terms of GOs, A + 0 = A is incorrect, unequal or inexact – however your POV chooses to define it. Remember that GOs are physical designs, therefore the “0” must be accounted for. For an identity to work contextually, the correct expression must be A + 0 = A + 0. It is possible that A + 0 = 0 + A, but the communitive property for GOs is speculation without understanding the designs they may take.

Contexts do not need to share all their dimensions to be added together. You may like 5 policy choices made by a political representative, but dislike 2 other choices. Here you are joined in 5 places and separated in two other places.

Because rules define levels, these shared dimensions may unite a Dominant Rule (DR) with a Recessive Rule in a complex context, or they may unite two Recessive Rules within a context defined by a DR. It may help you visualize this if you think of a complex context as a collection of dimensions organized in patterns controlled by rules. The stability of the context depends on the relationships of the rules that compose it – contexts are weakened and may break apart when internal RRs conflict with each other and/or with the DR.

Mixing is a method of addition that applies a rule to GOs not currently in a relationship. That rule is not an existing Dominant or Recessive rule in any of the GOs to which it is applied. That rule identifies at least one shared dimension in the construction of those GOs and creates a relationship between the GOs via that rule, and no other rule. “List everything in this room.” is an example of GOs connected by a DR that directly includes all the identical dimensions and indirectly all the non-identical dimensions of its component contexts.

Beyond the simple laws that are the foundations of math and logic – addition, subtraction, multiplication and division – there are laws that require more than one rule to build. These laws require more than one context for their uniqueness.

Inequality: A ≠ B (Identity, Not)

Irrationality: √2 (Square root, Infinite, Inexact)

Imaginary: I (Number, Inexact, Not on the number line, Not a real number)

Impossible: A / 0 (Division Law, Not on the number line, Inexact)

 

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