# Rudy's Diamond Strategies

This complementary Blog to the Chinese Challenge Blog is presenting studies to a mathematical theory of Diamonds. Diamond theory is studying for the first time, tabular categories as an interaction of categories and saltatories.

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## Sketches and exercises for dynamics and metamorphosis for formal systems

Abstract

The Ancient Chinese idea of a permanently changing world in which stable formulations, i.e. axioms in logic, are obsolete is thematized by the polycontextural strategy of permanently changing complexity. As a framework to realize complexity change for formal systems the kenomic matrix is involved. Examples for such formal notations are given and exercises to learn more about polycontextural diamond systems are proposed.

### 2. A remainder from Chinese Ontology

"Traveler, there are no path. Path are made by walking.” Antonio Machado

"A good mathematician is one who is good at expanding categories or kinds (tong lei)."

The Chinese philosopher Jinmei Yuan has given some crucial hints to the understanding of ancient Chinese mathematical thinking:

Chinese mathematical art aims to clarify practical problems by examining their relations; it puts problems and answers in a system of mutual relation--a yin-yang structure for all the things in a changing world. The mutual relations are determined by the lei (kind), which represents a group of associations, and the lei (kind) is determined by certain kinds of mutual relations.

"Chinese logicians in ancient times presupposed no fixed order in the world. Things are changing all the time. If this is true, then universal rules that aim to represent fixed order in the world for all time are not possible." (Jinmei Yuan)

An Aperçu
Chinese ontology (cosmology) can be put into two main statements:
A. Everything in the world is changing.
B. The world, in which everything is changing, doesn't change.
This two main statements are designing a paradoxical constellation.

Polycontexturality is complementing this ancient Chinese world model of harmony by dynamizing the concept of world-models:
C. A multitude of worlds are interplaying together.

The paradox to formulate mathematical rules in an ever changing world is very puzzling.
Many attempts to shed some light into it or even to solve the problem had been proposed.

It is not my intention to solve this ‘unsolvable’ problem.

Polycontextural logic attempts to formulate formal laws for an ever changing world. Nevertheless, we first have to abandon a Western interpretation of ‘change’. The Book of Change has nothing to do with Heraklit’s or Leibniz’s flux of things.

Many aspects about a philosophy of logic and time had been studied profoundly by the philosopher Gotthard Gunther. The connection of time and logic in polycontextural systems is not to confuse with any attempts of time or tense logics or physical time systems of any kind.

My own attempt to deal with the formal structure of changing first-order ontologies can be reduced, at this place, to two propositions:

Strategies of change
1. Diamond strategies: Each move is involved with its simultaneous counter-move.
2. Complexity strategies: Each move has to decide (elect/select) its intra-/trans-contextural continuation depending on the actual complexity encountered or created.

Because the strategies of change happens on the most fundamental levels of formal systems (logic, arithmetic, mathematics, ontology, semiotics, computability) a real combination of the antagonistic features of permanent change and formal operativity is opened up and accessible to realization.

One mechanism to realize change is given by the proemiality or chiasm between intra-contextural ‘parts’ and trans-contextural ‘whole’. A predicate defined inside a contexture can become the criteria for a new contexture which is augmenting the complexity of the contextural constellation.

For the sake of simplicity, 3 constellations of change are considered:
a) balanced constellation between formalism and application, with equal complexity for the formalism and the system to be formalized: compl(Form) = compl(System),
b) under-balanced constellation, with compl(Form) <= compl(System) and c) over-balanced constellation, with compl(Form) >= compl(System).

For classical Western thinking, based, shortly, on ontology and logic, only the balanced constellation with minimal complexity is available. Change is accessible in formal systems as change of complexion only. This strategy might be extremely sophisticated but it remains stable in respect to the logico-structural complexity of its paradigm.

Hence, not only every move (composition, concatenation, combination) in polycontextural diamond systems is accompanied by its hetero-morphic counter-movement but each movement is additionally determined by its polycontextural complexity-decision by election and selection.

In other words, in such a dynamic formalism, it easily can happen, that in the middle of a formal transformation (derivation, deduction, description, modeling) the complexity of the framework within those transformations happens might be changed, enlarged or reduced to legitimate a more reasonable and viable continuation of the transformations.

3.4. Exercises

### 5. Metamorphic changes

#### 5.1. Metamorphosis of topics

A transition from one contextural complexity to another doesn’t presuppose a pre-given existence of the new contextures. What might be presupposed is the possibility of change. And this possibility is realized by an application of the proemial mechanism between intra- and trans-contextural decisions.
An intra-contextural topic might become contextural prominence as a new contexture associated with the previous contextural constellation.

Reflection might change the meaning of an object by applying rules of chiastic metamorphosis.
Reflection is using the statement defining the object and this usage is defining the meaning of the object. Reflection and contemplation or introspection of an object can produce the insight that the meaning of the object under consideration is changing. Reflection as replication, thus, is augmenting the deepeness of the contextural complexity by a replicative, self-thematizing way. Reflection as iteration, is augmenting contextural complexity by an iterative, self-reproducing way. Alternatively, a reflection could change to an interactional augmentation of the contextural complexity. Both together, reflectional and interactional changes, are defining replicative, iterative and accretive contextural complexity of a polycontextural system.

The example below shows that the beginning reflection is interpreting an object as the number zero belonging to the topic numerals. This situation is implemented in a 1-contextural programming language. A second reflection considers the same object not as a numeral but as nil belonging to the topic of lists. Reflection has not to come to an end and can go further and with the interpretation and might realize that the object can be understood as belonging to the topic Booleans and appearing as the truth-value true.

Therefore the introduced syntactical object in its neutrality, observed and represented by an “external observer” in logis conceived as having simultaneously a numerical (in log), a symbolic (in log1.2) and a Boolean (in logmeaning. Hence, there is a chain of metamorphic replication from the topic Numerals, Lists to Booleans and a notation of the ‘neutral’ syntactic object “object” of Syntax. It starts with a reflection of the object “zero” of Numerals, ends with the Boolean “true” and gets a contextural abstraction as syntactic “object” in Syntax.

The example is designed for reflectional poly-topics in the experimental programming language ConTeXtures.