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ThinkArt Lab Animation: A.T. Kelemen© November 12, 1998 Dr. Rudolf Kaehr
Time is coming that we have to learn to live together at the same place without any chances of excluding each other.
Earlier on we solved this problem of living together with the help of the operation of separation and exclusion. Nobody had to live at the exact same place as someone else. The separation of two beings has given the space and possibility for interaction and cooperation between these entities. The separation was the fundamental condition for the possibility of interaction (cooperation, communication, co-creation, etc.).
Now it seems that we have reached the point that we have to develop a concept of living together in which we have to take place together simultaneously at the exact same place. It will turn out that this way of living together is prior to any separation and therefore to any form of interaction and cooperation. In classical terms two objects must be identical if they are not different. They are different if it is possible to separate them. How could togetherness be thought and conceptualized whithout the assumptions of identity and diversity and the procedures of identification and separation? How could this be possible? First of all, it isn´t possible at all on the premisses of the traditional concepts of place, object, state, separation and interaction. The reason is obvious, all these concepts are fundamentally rooted in the ontological and logical principle of identity.
In technical terms, how could it be possible that two different states of a computation could occupy the very same place in the computing space of their machine? Obviosly this is not possible at all. It isn´t possible neither from the point of view of the machine nor of the basic concepts of the programming languages. It is impossible for logical and physical reasons. Simply take the example of the definition of EQ in the programming langauge LISP:
EQxy =def if (eval x) = (eval y) then true
The equality EQ of x and y is strict, it is fullfilled or it is not - tertium non datur. The logic which is ruling these conditions is strictly binary. It is in whatever form a two-valued logical system which is ruling the conditions of equality. All in all, there are three levels of equality involved ruling this definition: the definitional (=def), the defined (EQ) and the defining (=). There is also no chance on the level of implementation on a more physical level of a machine. Two states are equal if they have the same address, and if they have the same address they have the identical physical realisation which is the equality =.
It seems that there is no chance to escape this situation.
Brian Smith has done a lot of work to clear and liberate this situation of strict ontological identity and bivalence in computer science. But at the end of his sophisticated work "On the Origin of Objects", MIT 1996 he introduces again a classical foundation for his quite liberal pluralistic and relativistic concept of truth and identity. It also doesn´t help much if we refere to our philosophers of the flux, Heraclit, Hegel, Whitehead, Deleuze/Guattari, Irigaray etc. because they don´t touch the topics of formalism and computation. The same is true for the more philosophical and theological work on togethernes by Heidegger, Buber, Binswanger, Levinas and others.
We will see, that togetherness in this study is not to be reduced to an anthropological category of, say Mitsein, Miteinandersein, Begegnung.
1 Kenogrammatical foundations of togetherness
Obviously we need a scriptural system which is beyond identity (of its signs and operations).
Everyone knows that the semiotical basis of any programming language is only possible because of the two fundamental operations of identification and separation of its signs.
Without separation there are no signs, and vice versa, without identification there is no separation. And without separation and identification there are no signs. And without signs there is no language. And so on...
Bad enough, my job has to be to develop a way of writing without or maybe even beyond the interlocking game of identification and separation.
How could this be possible? Try it with the idea of kenogrammatics!
There is not much but enough work done on this topic of kenogrammatics to understand at least the very idea of this quite radical exercise.
The procedure I propose works as follows:
1. Different from semiotics I introduce the concept of kenogrammatics and its morphograms with some definitions and rules.
2. A simple mapping of natural numbers onto the kenogrammatical system defines the trito-numbers of this kenogrammatics.
3. I introduce a decomposition of these trito-numbers into its binary parts. The result are the binary number systems with their rules of distribution and mediation based on the rules of kenogrammatics.
4. As a result we observe that the same physical" kenogrammatical address" gives simultaneously place to semiotically different objects (words).
1.1 A short introduction to the idea of kenogrammatics
Without jumping into the deepness of fundamental and foundational studies we can try to do a simple exercise.
In some sense we find in a stream of signs" a pattern and we are able to distinguish some features of it.
For short we introduce an arbitrary pattern MG with the feature (ABBCA),
First we say that this is not a list or a chain of signs, at least we have not to presuppose it. May be we simply don't know it now.
We call it a pattern. This means, that every other figure of the same structure is equivalent to this pattern. Patterns in kenogramatics are called morphograms. For convenience we accept the alphabetical order of the features for our notations.
MG = (ABBCA) = (BAACB) = (CAABC) etc. but also (*##+#) etc.
What can we do with such a morphogram? We can study it´s behavior!
Is the process of finding a pattern in a stream of signs not again involved in the same paradox of identification and separation as we know it from the previous introduction of signs? What is the difference?
As a metaphor think of a dominoes game where the dominoes are allowed to change size and the composition rule preserves the two-dimensionality in contrast to a Left-Associativ-Grammar (Hausser, 1989). The main principle is possible continuations in contrast to the well known principle of substitution. A crucial point of this principle is, as far as I understand it, that it doesn't appeal to a beginning (of recursion) and a pre-given alphabet of signs. It seems that this idea is more coalgebraistic than based on the concept of an algebra.
We will observe that the morphogram MG has the possibility to change. One form of change may be its growing. But this will happen in a strict way ruled by its intrinsic structure. Our morphogram may grow by repeating elements of its structure or it may grow in producing a new feature according to the underlying principle of possible continuations.
These are the only possibilities for the morphogram MG0 to grow.
Suppose we want MG0 to grow to the new morphogram with feature (ABBCAE). But this figure is equivalent to the pattern (ABBCAD). It produces nothing new in comparison to the MG4.
Another way of change can be realized without growing. The morphogram MG has the possibility to reduce or augment its internal structure between a minimum and a maximum of differentiation.
Reduction: (ABBCA) --> (ABBBA) --> (AABCA) -->... --> (AAAAA)
Differentiation: (ABBCA) --> (ABBCD) --> (ABCCD) --> (ABCDE)
It turns out that each morphogram we observe is constituted as an interaction of an evolutional and of an emanational procedure.
The rules of emanation are symmetrical in contrast to the asymmetry of the rules of evolution.
This is not in contradiction to the fact that all single morphograms can be produced or listed by the procedure of evolution alone.
1.1.1 Natural numbers in kenogrammatic systems
As a first step in our construction we can introduce a mapping procedure of the natural numbers onto the collection of morphograms.
MG4 = (ABBCAD) may become TZ = (122314)
Because the equality of two morphgrams depends only on its structure and not on its symbols the same happens for the numerical patterns.
The following exemples are trito-numerical equivalent (tzeq):
(122314) tzeq (23324) tzeq (322435)
1.1.2 Binary numbers in kenogrammatical number systems
We say that each kenogrammatical number TZ can be decomposed in several ways into its n-ary components. We will work with binary decomposition of trio-numbers.
Each pair of numbers i, j of a kenogrammatical trito-number TZ belongs to a binary number system (BTZ) defined by theses two numbers.
(122) is in S1 and (223) is in S2 but they are overlapping in the element (2). Also, (31) is in S3 and (1,4) is in S6 they overlap in the element (1) which belongs simultaneously to the binary system S3 and to the binary system S6.
We say composition and decomposition is restricted to the overlapping of only one element. The general case of overlapping with more than one element in common is called merging or fusion. Overlapping with no common element is called concatenation.
As a further example we use the operation of decomposition of a morphogram into its parts. The end and the start of two successive parts of two different binary number systems have a common element which is marking the jump from one system to the other.
Take the kenogrammatical number TZ= (0112000211002)
BTZ1 = 011/12/20/000/02/211/100/02
BTZ2 = 011/12/200002/211/100/02
Diagramm 1
Systemwechsel BTZ1
1.1.3 Chiasm of togetherness
What are the rules of the togetherness of binary structures (numbers) at the same place (locus) of a morphogramm?
We observe that we don't use the concepts of time and space to explain the behavior of our patterns.
Obviously kenogrammatic patterns, morphograms, are not ruled by the principle of semiotical identity.
The attribute of an object depends of its context, as what it is thematized or as what it figures. An object has in this sense no fixed attributes. The most general attributes of our objects are their beginning and their end or in another, but quite different wording, their head and their tail. At this point we are observing only the behavior of head (begin) and tail (end) to describe the functioning of the chiasm.
Diagramm 2
Diagramm 3
Chiasm of S1 und S2
Between head and tail we observe an order relation in the sense that first there is the head and then follows its tail.
What figures as head in one system occurs as tail in the other system and vice versa. They realize an exchange relation between the two concepts, head and tail in respect of the two levels S1 and S2.
Between the head (tail) of one system and the head (tail) of the other system we observe a relation of coincidence in the sense that both occurrence of the concepts belong to the same kind or category distributed over two levels.
To summarize: A chiasm between the concepts of head and tail is characterized by the relations of order, exchange and coincidence distributed over two loci.
We accept the constraint for this case that the tail of System S1 and the head of System S2 has only one element, therefore the exchange relation has only one element in common in contrast to the common definition. With that we have an interlocking mechanism of the composition of objects of different levels.
1.2 Interpretations and specifications
Diagramm 4
TZ= (01120211002)
1.2.1 Dekompositionen
Die als Trito-Zahl notierte Ereignisfolge TZ im Gewebe dreier Binärsysteme S1, S2 und S3 mit den 3 Elementen {0, 1, 2}. Je 2 Elemente definieren ein Binärsystem.
lässt mindestens zwei Deutungen zu:
a) 011/12/202/211/100/02 mit der Systemfolge: S1S2S3S2S1S3
b) 011/112/202/211/1100/002 mit Systemfolge: S1S2S3S2S1S3
Hier ist zwar die Subsystemfolge der beiden Auflösungen die gleiche, die Auflösungen selbst sind jedoch verschieden in ihrer jeweiligen Länge.
Die Trito-Zahl TZ= (0112000211002)
lässt Deutungen zu, die sowohl die Subsystemfolge als auch die Länge der Subsystemfolgen betreffen.
a) 01/12/20/000/02/211/100/02 mit S1S2S3S1S3S2S1S3, l=8
b) 01/12/200002/211/100/02 mit S1S2S3S2S1S3, l=6
this number can be produced by two different successor sequences
##the order has to be inverted####
a) suc1(suc2(suc2(suc3(suc3(suc3(suc2(suc2(suc1(suc1(suc1(suc3(suc3)...(zero1)...)
011/12/202/211/100/02 with the subsystem sequence: S1S2S3S2S1S3
b) suc1(suc1(suc2(suc2(suc2(suc3(suc3(suc3(suc2(suc2(suc2(suc1(suc1(suc1(suc1(suc3(suc3(suc3)...(zero1)...)
011/112/202/211/1100/002 with the subsystem sequence: S1S2S3S2S1S3
Different interprations of a poly-numerical event
"The law which we applied was the principle of numerical induction; and although nobody has ever counted up to 101000, or ever will, we know perfectly well that it would be the height of absurdity to assume that our law will stop being valid at the quoted number and start working again at 1010000.
We know this with absolute certainity because we are aware of the fact that the principle of induction is nothing but an expression of the reflective procedure our consciousness employs in order to become aware of a sequence of numbers. The breaking down of the law even for one single number out of the infinity would mean there is no numerical consciousness at all!" Cybernetic Ontology, p. 360
Diagramm 5
I Systemwechsel mit Bifurkation
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