Rudolf Kaehr Dr.^{@}

ThinkArt Lab Glasgow

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.

It is said, that categories are distributed over the kenomic matrix.

As pointed out, dissemination is composed of distribution and mediation. How is mediation working? Again, different contextures are mediated by their proemial relationship. That is, by the proemiality of their basic terms. What are the basic terms of categories? At first, it is mentioned that categories are build as compositions of morphisms. And morphisms are mappings with domains and codomains,conceived as objects. Hence, composition of two morphisms is ruled by the matching conditions of the codomains and domains of morphisms.

With such a proposition of the scenario, the pre-conditions of categories are not reflected. That is, their locatedness is left to the mental activities of the categorist.

**How to find a mediation?**Therefore, the foremost step of distributed category theories, and not only distributed single categories, is to find possibility to encounter category theories at other loci in the kenomic grid, able to get into an interaction and getting mediated. On such a path, i.e. journey, categories might occur which are structurally not prepared for mediation. Hence, strategies have to be developed to find the adequate setting of categories or contextures and reflectional techniques might be applied to adapt automatically to the situation.

Most attempts to interact will fail. Some will succeed only partially, some, probably small systems, will succeed totally. All levels of possible interactions have to be accepted and studied.

Such a selection to find a winning mediation is not unfamiliar in the theory of formal systems. Formal systems are build on a

**Mediators***"A new kind of interpreters appears to the programmer in ConTeXtures, the *mediator*-*interpreter*. This kind of interpreter has to collect, control and to establish the mediation of different programs written by different programers at different locations at different times. **In contrast to existing compilers and interpreters with their hierarchic tectonics, the new situation can be defined as *heterarchic

Today, such a societal collaboration might be called Web 2.0 selection, compilation and interpretation. It might help to design the idea of a *heterarchic* Web Computing Paradigm which is not reduced to data or service sharing (cloud and grid computing). *Mediation* is not sharing but *creation*. Hence, a collective system production is not a collective sharing system but should be conceived as a genuine societal computing paradigm. This is not only intended to surpassing the deadly anachronism of Big Corporations but also the individualistic limitations of Open Source strategies.

*"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.

2.

Intra-contextural continuation is supposing that the logic-structural complexity (of logic, arithmetic, semiotics, ontology) is stable and hasn’t changed, hence

Trans-contextural continuation has to reflect the possible change in complexity and has to chose, i.e.

In classical arithmetics, the step from n to n+1 is unambiguously defined by the arithmetical rules or axioms. In contrast, polycontextural arithmetics is involved always, in at least, two actions, election and addition, producing a kind of a 2-dimensional tabular continuation:

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.

All answers to the exercises can be written in English, German or French and posted to my Blogs. Chinese and Japanese proposals are welcomed.

The kenomic matrix was introduced in ConTeXtures to offer a general notational approach to the dissemination of formal systems (logics, programming languages, semiotics) considering the modi of dissemination (identity, permutation, replication, iteration, bifurcation) as strategies to implement computability, reflectionability and interactionability into formal systems.

The term “kenomic” refers to ‘keno’, greek for empty. The matrices should be read as empty of logical and mathematical presumptions. That is, their mathematical features are not considered as important for the definition of the dissemination of formal systems. Such a reflection on the epistemological status the matrices would deserve an own contemplation. Metaphorically, matrices are not more than the naked shell of the turtle in the story of Lo Shu.

Hence, the kenomic matrix consists of empty places which might be occupied by formal systems or not. Like in kenogrammatics, kenoms are not linearly ordered but are inscribed in a tabular manner.

This openness allows to interpret kenomic matrixes naturally for reflectional and interactional constellations.

Regarding the sketched patterns for kenomic matrices as general place-holders for formal systems, applications for the distribution of the syntax of specific systems are following naturally.

Distribution of logical connectors and quantifiers and their complex variables are constructed along the frame of the involved matrices.

As an example of the use of the matrix approach for composed formulas, the first-order formula for categorical composition might be involved.

The short version of the formula in a 3-contextural situation, involving a transactional quantifier Q, too, is given below. Such short versions, presented usually in a Guntherian context, are working only for very simple cases and are mostly misleading.

Hence, we have to take the burden and offer an explicit notation form of the formula. It is easy to understand the distribution of all elements involved: *variables* x, y, *predicates* C, D, K, *quantifiers* , , Q. The distribution of equality (=) is omitted.

This is not the place to go in deeper details of polycontextural syntactic notational systems. It is easily to see that most of the combinatorial possibilities are not well-balanced. Again, such situations are not unusual, they appear in a much simpler combinatorics in classical formal systems too.

Hence, e.g., a logical constellation as for *table-a,* with complexity (3,3), can be changed to different tables depending on the type of the required complexity and complication with complexity (3, 4) , (4,3) and (4,4). Such changes are involving the formal systems as a whole. An example for a change of complexity concerning the *quantifiers* only of polycontextural logics is given below.

Change for polycontextural systems has many faces. Additional to accretive extensions, a system might extend its scope by reflection into itself, self-reflection and introspection. This gets a formal representation by the super-operators *iter*, for iteration and *repl*, for replication.

- *Iteration* is augmenting complexity iteratively from SS

- *Replication* is deepening complexity without augmentation from SS

In the example, system Sat (O (O, leading to S and S

**Interplay between interactionality and reflectionality**Mixing freely reflectional and interactional pattern are leading to local iterations and recursions of the general scheme producing a fractalization of the general scheme.

Hence, the whole reflectional/interactional pattern of the example is: [G

**Interplay between interactionality, reflectionality and replicativity**Additional to the example above for interactionality and reflectionality, a pattern of

**Permutations**

Permutative patterns, produced by the super-operator *perm*, are behind those visits to other systems and back to the start again. The journey might start simultaneously in system_{1} and system

The table represents more the static pattern, while the bracket notation the dynamics of this permutation.

Recommended articles:

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 *log*is conceived as having simultaneously a *numerical *(in* log*), a *symbolic *(in* log*_{1.2}) and a *Boolean* (in *log*meaning. 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*.

As the example shows, the reflectional distribution of the topics Number, List, Boolean, building the category “poly-topics” , is introduced as (*zero, nil, false*) at the locus O_{1}. *Thematize (zero, nil, false)* is distributed reflectionally over 3 places by the super-operator *replication* (repl) and neutrally represented by the syntactical object “object” at the place O_{3}. In this case, the positions at place O

*Exchange relations:*- "

as: define

- "

as: define zero from contexture1.1 as

- "

as: define nil from contexture1.2 as

This change of identity of the topics from one contexture to another by reflection/replication is producing a

Thus,