Polycontextural and diamond dynamics

Sketches and exercises for dynamics and metamorphosis for formal systems

Rudolf Kaehr Dr.@

ThinkArt Lab Glasgow



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.

1.  Mediation in complex dynamic formal systems

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 syntactic selection, cut, between correct and non-correctly composed sign sequences. Here too, most sign sequences which are possible as combinations of the signs of the pre-given sign set (repertoire, alphabet) are not accepted as formulas. Hence, the accepted formulas are a small subset of the free monoid over the alphabet.

"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. Heterarchic compilers/interpreters have to manage the mediation of the different hierarchic approaches of programming. This concept of heterarchic compilers opens up a new kind of societal collaborations.” (Kaehr, ConTeXtures, 2005)

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.

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.

Intra-contextural continuation is supposing that the logic-structural complexity (of logic, arithmetic, semiotics, ontology) is stable and hasn’t changed, hence selects its next step.
Trans-contextural continuation has to reflect the possible change in complexity and has to chose, i.e. elect its contextures, i.e. its contextural environment for its next steps to select.
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:

typeset structure

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.

2.1.  Exercises

2.1.1.  Collect arguments - pros and cons, and beyond- and articles given in my Blog and elsewhere, which might support or reject the ‘Apercu’ of a Chinese Ontology and a Diamond World Model.
2.1.2.  How are those thoughts connected to the project of Derrida’s Grammatology and the deconstruction of phono-logo-centrism in formal systems? Read and comment original texts only (if necessary translations)!
2.1.3.  What can you learn from the sketches to a new rationality based on polycontexturality and the concept of Chinese scriptural paradigm for the understanding of the decline of the Western Hegemony?
2.1.4.  What are the immanent limits of Western thinking and how might they influence the economic and financial crash? Connect your insights with the proposals given in my “The Logic of Bailout Strategies".
2.1.5.  Create more questions and answers of this kind.
2.1.6.  A good exercise to experience the patterns and strategies of polycontextural and diamond thinking for more familiar topics, like ethics, human rights, identity, pluricentrism, Web 2.0 etc. might be the reading of the ‘exercises’ I have written in the collection “Short Studies 2008".
All answers to the exercises can be written in English, German or French and posted to my Blogs. Chinese and Japanese proposals are welcomed.

3.  Notational notes

3.1.  The kenomic matrix

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.

3.1.1.  Computational, reflectional and interactional constellations

An empty kenomic matrix with m = 6, n = 3 : [O   O   O]                                        ...                                3    3.1                         -                             3.3

3.1.2.  Dissemination of logical particles  

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.

3.2.  Balanced formulas

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

(C3 ) :   mono - contextural composition <br /> ∀ x ∀ y ((∃ z : K (x, y, z) ≡ (C(x) = D(y))) <br />

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.

(C3 ') : 3 - contextural ,   short <br /> (Q^1 ∀ ^2 ∀ ^3)  ... 704; ^2 = [∀ ^2.2],    ∀ ^3 = [∀ ^3.3]

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 typeset structure, typeset structure, Q. The distribution of equality (=) is omitted.

(C3 '') : 3 - contextural ,   explicit <br /> [         1.1                                    ...               D      -              D      y      -              y

<br /> Formula notation <br /> short : <br /> [bif _ 1, id _ 2, id _ 3 ...                               3      3.1                       x                               3.3

<br /> Bracket   notation   for   [ bif, id, id ]

[[                                                                                             ...            (G   )                                             003

3.3.  Non-balanced formulas

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.

<br /> [                1.1                                            ] ((         1.1        ...     x                           -                                   x    ∨ y

3.4.  Exercises

3.4.1.  Write an overview of typical notational constellations for balanced formulas. Use the sketches given in ConTeXtures and From Ruby to Rudy.
3.4.2.  Program features of balanced (m,n)-contextural notational systems for junctional, transjunctional connectors and quantifiers.
3.4.3.  Try to define and program more efficient and ‘ergonomic’ notational approaches to general tabular syntactics.

4.  Sketch for complexity changes

4.1.  Extending constellations by accretion

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.

 Table - a <br />           1                 2                 3    O                 ...               x                                             3.4                                4.4

Table - b <br />                              O                    O           ...       4                              4.1                          2.4                          3.4

<br />  Example of complexity elections for quantifiers <br />                                 ...                 Q             ∃       ∀       ∃ 

<br />  Overscript[==>,      elect (4, 3)     ] int ...                ∃                            ∀     ∃    )

4.2.  Extending constellations by iterations and replications

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 Stypeset structureStypeset structure
- Replication is deepening complexity without augmentation from Stypeset structureStypeset structure

In the example, system Stypeset structureat (Otypeset structure (Otypeset structure, leading to Stypeset structure and Stypeset structure

FormBox[RowBox[{RowBox[{<br /> Formula notation <br /> PM = ( id _ 1(rep _ 1 r ...                                                                   3   ∅       2            3

4.2.1.  Changes from 3x3-diagonal matrix and bracket to reflectional/replicational and interactional constellations

Table and bracket notation for diagonal 3 x3 - matrix FormBox[RowBox[{ , RowBox[{FrameBox[Cell ...                                 Bracket notation for reflectional change from 3 x3 to 5 x3 matrix

FormBox[RowBox[{[[                                                                             ...                                                                                              03303

Bracket notation for reflectional/replicational change for 5 x3 [[                             ...                                               5       5.1 .1 .1 .1   -                        5.3

<br /> Alternative notation for reflectional/replicational change for 5 x3 <br /> [[           ...                                                       22200                                  03303

<br /> Bracket notation for interactional change for 3 x3 <br /> [[                            ...                                                                                                003

Bracket notation for interactional and reflectional/replicational change to 3 x3 <br /> [[     ...                            020                                                                 003

Bracket notation for interactional and reflectional/replicational change to 4 x5 <br />          

                                         O                                O    ...     -                                                 4.4                                      4.5

   [                                                                                 ...                                                                                               0505

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.
The examples shows:
At the locus O2 we have a full reflection G222 and an interaction from the locus O1 into the locus O2 producing additionally to G222 at Otypeset structure and an interaction from the locus O3 into the locus O2 producing the interactional pattern G003.
Hence, the whole reflectional/interactional pattern of the example is: [G111, Gtypeset structure, G033].

          [                                       ...                                                                                                033

Interplay between interactionality, reflectionality and replicativity
Additional to the example above for interactionality and reflectionality, a pattern of replicativity or introspection is involved at O1with Gtypeset structureand Gtypeset structure

[ [                                                                                            ...           (G   )                                              033

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 system1 and systemtypeset structure
The table represents more the static pattern, while the bracket notation the dynamics of this permutation.

[[                                                                                           ] ...                                                                                                123

4.3.  Exercises

4.3.1.  Collect the arguments and constructions given in my articles and build a systematic model of the dynamic interplay of interactionality/reflectionality and interventionality in formal systems.
Recommended articles: ConTeXtures. Programming Dynamic Complexity, Godel’s Games, Actors and Objects, From Ruby to Rudy, How to compose?
4.3.2.  Compare those polycontextural and diamond models with models from modal logic, cognitive science, theory of reflection (Levebvre), reflectional programming (Smith, Maes) - and others.
4.3.3.  Play around with your own ideas. Would it make fun to simulate polycontextural diamond dynamics with cellular automata models? What could we learn from such modeling, simulation and implementation? What would be lost?
4.3.4.  Dynamics based on the ‘kenomic matrix’ might be studied for logical, arithmetical, categorical and semiotic systems by applying the materials proposed by now.
4.3.5.  What are the structural consequences of contextural change for diamond category theory?

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 logtypeset structureis conceived as having simultaneously a numerical (in logtypeset structure), a symbolic (in log1.2) and a Boolean (in logtypeset structuremeaning. 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.

[                                                                                              ...                  a

Table notation                  O     Overscript[==>, &nbs ...                      3      1.3                       x                                       3.3

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 O1. 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 O3. In this case, the positions at place Otypeset structure

Exchange relations:
- "define zero" is "define zero as zero", as the start of the levels.
    as: define zero in contexture1.1 as zero in contexture1.1
- "define nil" is "define zero as nil",
    as: define zero from contexture1.1 as nil in contexture1.2
- "define false" is "define nil as false".
   as: define nil from contexture1.2 as false in contexture1.3.

This change of identity of the topics from one contexture to another by reflection/replication is producing a chiastic chain guaranteeing the connectedness of the step-wise reflection of the whole. Levels and meta-levels of reflection are connected by means of proemiality realizing its structural rules of exchange, order and categorial correctness (coincidence).
Thus, define name is an abbreviation for "define namei as namej" with i=j.

5.2.  Exercises

5.2.1.  Construct examples for reflectional, interactional and interventional constellations for poly-topics in the framework of ConTeXtures.
5.2.2.  Construct further examples in the framework of ConTeXtures with topics like semiotics, logic, arithmetics.
5.2.3.  Describe ‘empirical’ situations where such contextural changes of augmenting or reducing complexity seems to be unavoidable.
5.2.4.  Try to develop a polycontextural measure for complexity.