| |
|
|
|
ThinkArt Lab Animation: A.T. KelemenĐ November 12, 1998 Dr. Rudolf Kaehr
Does the transjunctional super-operator BIF exist for bool0(3) ?
Booleans are atomic therefore they can not be split by the super-operator BIF. But this is true only under the condition that they are ruled by the law of identity. Why should they?
Cloned objects are the same but they are not identical.
How can we replicate systems properly if we cannot replicate their atomic terms? Dissemination as distribution and mediation, or as replication, gives us no answer about the splitting of atomic terms because the distributed systems are identical for themselves, e.g. bool1, bool2 and bool3 are all containing and conserving the classical elements, values and operations in their domain. Modeling "as such", "as other", ...
Normally we know transjunctional operations in polycontextural logics only from binary logical operations, the so called transjunctions. Bifurcations are generalizations of this binary concept to super-operators which are ruling poly-contexturality as such and not only internal or local operations. Also super-operators are well know in the polycontextural literature they have never been studied independently from their historical sources. In this sense the question of a generalized form of bifurcation, applicable to all objects of a system, was not focussed. Here I try to develop some ideas and constructions of these generalized bifurcations with the help of the metaphors of cloning, replication and not only by the metaphor of splitting, parallelism, simultaneity. In some sense, we even can say, that bifurcation, simultaneity etc. are cases of cloning. More explicite, these terms seems to form a system of complementarity. There is no bifurcation without replication-and vice versa. A free use of the idea of replication goes beyond the well known transjunctions in polycontextural logics. I can not go deeper into the development and explication of the metaphor cloning", but it has to be mentioned, for short, that the very idea of logical cloning and replication as well as of logical bifurcation and simultaneity is based on the kenogrammatic concept of morphograms. Morphograms are structural patterns invariant to logical negations, therefore byond identity and diversity of signs, which means outside the realm of signs.
From a more technical point of view I am abandoning in a further step the basic functional approach of the historical polycontextural logics based on a interpretation of multiple-valued logics. One of the earlier significant steps was to abandon the Cartesian product approach of n-ary functions and their problem of decomposition into two-valued subsystems (theory of place-value systems).
Example (ID, BIF, ID)(opn1, opn2, opn3)
ID1 T, F --> bool01 : {T1, F1} simul {T2, F2}
BIF1 T, F --> bool02 .simul. bool01 : {T2, F2}
ID3 T, F --> bool03 : {T3, F3}
BIF1 bool02 --> bool02 .simul. bool01 --> bool01
BIF1 bool02 x bool02 --> bool02 .simul. bool01x bool01 --> bool01
ID3 bool03 x bool03 --> bool03
Diagramm 1
(1, 2, 3)-->(12§23§13)-->((12%23)§23§13) :: ((12%23%#)§(#%23%#)§ (#%#%13))
If we apply the unary operations of negation to this evaluation (mapping) (Belegung) we have to deal with the quite new situation of a reflection of the values of system2 in system1. It seems to be reasonable to accept that a negation of the values in system2 has to be mirrored one-to-one in system1 to be correct. We say that system1 has a model of the behavior of system2 in itself. The model is not the original, it differs in its place in the reflectional system. If system2 changes its state the mirrored model in system1 has to change in exactly the same way. Does this make any sense? Probably it is the most simple case of transjunctional or replicational distributions.
1 Bifurcational distribution of negations and junctions
If we start with bifurcation we are forced to distribute all operations in a transjunctional way. This is really a new and intriguing situation. Not only we have to involve constants, negations but also binary junctional operations like conjunction, disjunction into this transjunctional game. And the originary transjunctions of the old place-value system of logic are understood as a very special but quit explicit case of bifurcation.
Does it make any sense to repeat exactly the same logical situation of one system in another logical system which is distinguished from the first only by its different place in the complexion of the whole polycontextural logic?
From the point-view of a theory of argumentation (interaction or communication) this type of modeling corresponds to the situation when an actor is agreeing in all logical points with its partner of communication. The agreeing system has its own position and its own logical arguments but additionally it offers space to the other system to accept its logical arguments. Insofar the modeling has to be strictly one-to-one. To reduce the situation of agreement to the usual case the actor denies its space and accepts the arguments in the space of the other system. I agree, but keep it for yourself. There is no logical space I can offer you for that."
From the point of view of the model of reflectional programming this situation of mirroring the logical constellation of the environmental or partner system could be understood as an interpretation and modeling of the so-called causal relation" of a reflectional system on a logical, and not on an informational, level.
This is the special case of transjunctional behavior. The general case accepts a different logical behavior, a different sequence of argumentational steps at the locus of the accepting system. This is the real case for logical transjunctions as we know them from polycontextural logic.
In this new context the operation of bifurcation is distributing total functions and not partial functions as it is necessary for transjunctional operations. To distinguish the two concepts, this type of function should be called replicative transjunction or simply replication (of functions).
We also have to consider the case, that combination of logics has not to be homogeneous, that means, that we are mixing different types of logical systems together. Therefore the bifurcation operation of this different systems produces inside" of reflecting system a mapping of components which are from a different type of logic than the reflecting system itself. But this possibility is out of the range of this study, which is mainly introductory.
1.1 Problems of the beginning and the beginning of problems
1.1.1 The beginning as zero
Additionally to the successor operation letīs introduce the inverse function of the predecessor pred.
As we can see, our natural system introduces zero as an absolute beginning. There are no predessors of zero. And there is no number x with a succession to zero. That is, the function pred(x) is not defined for x = zero.
More explicit: nonEx(x): suc(x) = 0
The term zero seems to be a very priviledged object. It is the beginning of everything, in this sense it is not only a beginning of many other beginnings, but an origin. It is called an initial object. And later we can show that there is one and only one such initial object, all others are strictly isomorphic to it. The whole richness of the pluralities of beginnings is reduced to the general and abstract initial object as the only origin. Plurality is possible only in a secundary sense of applied, that is concrete or even empirical systems.
And this is exactly how it has to be for human beings. There is one and only one beginning-and this beginning is the origin of everything.
Ask Aristotle why it has to be this way.
And knowbody should think that there has been the slightest change in this mono-contextural archeology since Aristotle.
2 Are there any neighbors of zero?
All that is in sharp contrast to my construction of a plurality of natural systems. It also violates my principle, that there are many beginnings but no single origin.
Because I accept that locally for each natural system, zero is an initial object and therefore there is no predecessor of it, I have to introduce another wording. Despite the fact that the initial object has no predecessor it is more natural to speak of many neighbors. In other words, zero has predecessors but not in its own system but in its neighbor systems, therefore these predecessors are strictly speaking neighbors, that is neighbored initial objects.
And therefore in the strict sense, but maybe broken down, split, of the meaning of the term initial", there is no initial object left at all. Initiality occurs as cloned and dispersed in plurality. If there are many initial objects, the notion of initiality is changed and has lost its init and its unicity, guarantied, before deconstruction, by isomorphism.
neighbi(neighbi(x)) = x, i=1,2,3
neighbi(neighbj(neighbi(x))) = neighbj(neighbi(neighbj(x))), I,j = 1,2
For 4 contextures, we have additionally to the cyclical equations a commutative eqation:
neighb1(neighb3(x)) = neighb3(neighb1(x))
This approach to polycontextural arithmetics is still very static and presupposes that there are something like pre-given arithmetical objects and orders between these objects. It suggests, that the object zero" is a stable arithmetical entity.
A more dynamic approach is developed if we remember that we are much more focussed on the operations, and their operationality, than on their objects. In this sense zero is not an object but a function or operation which can be realized independently of a special object. As in the classical approach each object chosen as a beginning is considered as isomorphic to zero. But this is considered on the level of the models of the abstract system of natural numbers or the abstract word algebra. It is a model theoretic consideration and does not belong to the level of the abstract system itself.
In polycontextural system we rencontre a very different situation. Because we have a multitude of abstract systems there are complex possibilities of interaction between these abstract systems without leaving their abstractness for the purpose of modeling.
In the case of the static approach we have only the possibility of reaching the different zeros from a zero in a given system. That is, the zero of a neighboring system is reached as the neighbor of zero in a chosen system. Functions which are not zero do not have a neighbor in another system which is a zero function.
This statical situation is radically changed in a dynamical system. Each function can have its own zero neighbors. Arithmetically speaking each number in one system can change its functionality to a beginning in another system. And each beginning in one system can be an ending in another system. Therefore, there is no absolute beginning needed, and an ending has not to be connoted with attributes like potential or actual or factual or whatever type of infinity nor with the concept of finity. All this Greek heritage will be in the play in a much later step of arithmetical thinking.
Maybe we have to distinguish between the description of an arithmetical process, which leads to the well known formal systems, from Peano to Lorenzen, and the notion, the conceptualisation of an arithmetical system. The description tells us what natural numbers are doing, or better, what happens if we use numbers. This leads automatically to the problem of antropological problems of the limits of usage of numbers or in contrast to the non-antropological, but Platonist concept of usage or existence of numbers.
But there are other ways of thinking, too. It has its occurrence in Hegels Logic and its further development in Gunthers Natural Numbers in Trans-Classic Systems.
As I have shown before, the idea of proemiality is to inscribe the difference which constitutes all relations and operations as such. Proemiality is the prelude to all operations in formal systems. It is the constitution of all institutions as formal systems. A trans-classical approach to the problems of introducing natural number systems is therefore to apply the proemial relation, that is the strategy of chiasm, onto the arithmetical system. I have to suspend the questioning of the totality of the term "all" in these statements.
This leads to a characterization of a dynamical approach written, inscribed, as a chiasm between the four terms: inital, final, successor, predecessor.
This chiasm or proemial relation between initial and final, successor and predecessor, does not need a fixed beginning, it doesn't force us to accept a decisionist beginning or start of the abstract system by a privileged initial element, written as a introductory rule of level zero. On this level, there is also no need to be concerned about infinities of all sorts.
On the other hand, it offers a mechanism for a mediating interplay of cognitive and volitive structures and actions in a formal system.
Further, this chiasm makes it reasonable to speak of obstacles in arithmetical systems without being confused with the problem of the existence or nonexistence of numerical objects which should be and could not be the last number in a series of numbers.
As much as the first number is relative, as much the last number is relative, too. Beginnings and endings are an interacting couple of terms, and not an asymmetrical one, in the sense of one beginning and no end as in classical arithmetics.
After having introduced this idea of a chiastic interplay of the primary terms of the trans-classical concept of a plurality of arithmetics, it is not unnatural to specify a special case of this dynamics to define a very special statical system, the system of natural number series as we know it.
3 Deconstructing the origin
It is mentioned in philosophy by Derrida that a sign or a mark has to be able to be repeated, iterated, otherwise it cannot be a sign or a mark. It is the iterability of signs and marks which makes them a mark or a sign. Without going into the highly complex work of deconstructing the origin, the initiality, the initiality of the origin, of zero and other logocentric concepts I like to mention that the whole problematics of the origin is reassembled in the rule or function of introducing, or postulating, zero.
It can be mentioned that the zero can be repeated by the successor function. But this is not exactly the case, what is repeated is the successor function applied on the single and only object zero. This is clear if we reconsider that the suc function is of unary type and the zero is of arity-zero. It also can be mentioned from a semiotical point of view that the sign zero" can be repeated as often as we wish and that therefore zero" is obviously a sign. But repeatability shouldn't be an abstract concept. Iterability happens in the context of a defined system. And here I am thematizing the concept of an abstract natural object, that is a word algebra with its initial object zero.
In this sense, and not in a arbitrary other sense, it is the case that the fact that zero as the only start of the word algebra has no predecessor, means that this object zero" is not repeated, cannot be repeated and isn't allowed to be repeated. Zero is not an event of iterability. Iterability of a sign doesn't mean that the sign has to be iterated, but that the sign has the necessary possibility of being repeated.Obviously, the semiotic status of zero as an initial object in the word algebra, and similar in all other formal systems, is very unclear.
The decision to make a start with zero is not inscribed in the formal system which contains zero. It could be said, that the introduction of zero as the initial object can be understood as a one-step iteration. Iterability can be introducing and repeating. But this is in strict conflict with the concept of an initial object, insofar, as initiality is the start of repetition, and what is the starting point of repetition is not itself, in the same sense, repetition, too.
In this consideration, zero is the initial object of the word algebra and not the number zero". We have not to be confused with possible interpretations of the number zero as ontological notions like nothingness, or emptiness or as arithmetical place-holder for numerals in a positional system and so on. Zero here is simply the mark of a start. It could be a stroke in a stroke calculus with the initial object stroke" introduced by the arity-zero rule: introduce a stroke! Or, it could by a cross in the calculus of indication: draw a distinction! Mark it!
Nearly all philosophical or meta-mathematical studies are concerned about the problem of infinities of all sorts, later they consider the problem of finity, too. But there are no studies about the strange situation of their initial object, zero. Sometimes zero is thematized as an ontological problem, mixed with notions like nothing, nothingness or even emptiness. But the very character of the initial object to be the initial of the series of the, say, natural numbers, is not worth a reflection. The reason may be very simple, it makes obviously no sense at all to think of the initial, the zero, as a derivate of something else.
And by the way, Aristotle has cleared in his attacks against Pythagore and Plato the scene for ever.
I think that we are in the hypnosis of some quite odd or at least strange dichotomies. The Greeks had been very careful with the problem of the beginning of the series of natural numbers. Aristotle organized a lot of highly intriguing thoughts in defending his initial object, arche. The queer distinction they introduced is the dichotomy of beginning/infinity. Because Aristotle had to fight against mythological circular reasoning, he couldenīt introduce the more harmonic dichotomy of beginning/ending, arche/telos.
They also didn't introduce the dichotomy of zero/ininity as an interpretation of nothing vs. all, simply because the didn't work with the concept of zero.
Surely, the character of the inital object changed in history from the one (1) to the zero, but the pattern of the main questions remained. The questions, from theological, philosophical and mathematical point of view, had always been about the concept of infinity in all its disguises. Bad enough, all the work was done as a family affair between Aristotle and Plato. Some criticized everything which is not finite as bad platonism, the other ones wanted much more than infinities, but trans-finities and even more. And today we are criticizing the concept potential infinity in the name of a more terrestrial and anthropological concept of finity, which should fit much better with the scenario of finitarism in todays computer science.
Even my own hero Aleksander Yessenin-Volpin is more concerned about all his finities than with any deconstruction of the inital objects of his NNNS (Natural Number Notational Series).
4 Brian Rotmanīs attack
Although Brian Rotman supports the idea that numbers are made by counting, and are not to be presuposed as platonic entities, an idea which is well known by constructivists (Bishop, Lorenzen), he is not aware about the problem of starting to count. As it is well known, no formal system starts to work, to draw conclusions, to construct numbers, and so on, without a decision of a user to start the system. And this decision has no representation in the system itself. It is not included that a formal system has to realize its one start.
All these approaches are Platonist in the sense that they are primarily cognitivist. They deal only with the cognition of these processes and procedures. All volitive decisions and actions are excluded not only from the formal systems, but more important, from all rationality at all. Earlier on there was a big confusion between psychological, logical and even neurological aspects of logics. After Husserls big attack against psychologism in arithmetics and logics there was a new confusion which lead to the tabu of considering reflectional aspects of formal systems because of the fear of psychologism. Today, there is nearly no limit of confusing every thing with every thing.
From a strict formalist view-point the idea of an interacting mechanism of cognition and volition for formal systems is beyond any rational discourse and is strictly excluded from the academic world.
The other obvious restriction of the arguments of Rotman, and probably all other philosophers of mathematics and the mathematicians themselves, is simply the fact, that he is dealing with the numbers, in what ever onto-semiotical status, as objects. His concerns are the numbers and not the processe, or the processuality of counting in itself. The results of the process of counting are his objects under consideration and not the very process which is producing his objects. All in all, it is some productionism which determines his idea of incorporated numbers. But we have to accept, that this approach is very natural, because we have to use sings as objects and we donīt have any help to write a process as a process, there is no notational system for processes as such. In a process algebra, we are dealing with the names of processes and not with the processuality of the events. Similar in a musical notational systems, the notes are fixed, they tell the musician what to do, which process has to be started. In this sense they are a notational fixation of musical thoughts, in other words, they are signs or markers and ruled by the semiotical concept of identity. More explicite, it has to be mentioned, that the whole adventure of mathematics and philosophy is written in the framework of writing, even in the world view of the book. Maybe it is a principally absurd and totally Sysiphus-type of work to try to do something else than to write.
The strategy of Rotmanīs intervention is to cut down the phantasies of Platon and Cantor in reducing the possibilities of repetition.
"One needs to see how and for what reasons one can refuse the idea of perfect repeatability; ..." , p. 54
From the point of view of thematizing the metaphorical arsenal of his argumentation, it is obvious, that he is in the metaphorizity of the steps, the stepwise counting, with and whithout restrictions. In contrast to Yessenin-Volpin, there is nothing to see like metaphors of jumping, switching, leaving...the series of natural numbers for other, neighboring, series of natural numbers.
"Counting , ..., is an activity involving signs. And, as an activity, counting works through-it is-significant repetition." p. 6
Who is counting? Is it not, again, a transcentental ego, disguised as a singular empirical anthropological subject, which is counting, all these numbers in his or her solitude? Why not an interacting multitude of counting agents?
As I argued, the start of the whole story, demands for a decisionist introduction of the inital object, zero. It turns out that zero as an arity-zero term is not a sign. It canīt be a sign in the proper sense because it is a non iterable term. This term is postulated, but not repeated in its word algebra. Again, what is repeated, and what can be repeated, is the arity-one term suc, the successor function, which, by its name means what it does. The suc-function is repeating itself applied on the initial term zero. As a result of this repeated application we get a series of repeated zeros. But again, this plurality of zeros is not a plurality of the initiality of the initial object zero, but a result of the application of the second rule, the suc-function and not a repetition of the first rule, the introduction of the initial object zero. The first rule will never be repeated.
Again, I am obviously not speaking about the impossibility of repeating the word "zero", because, as we can see, I am repeating this word "zero" quit a lot in my text. The word "zero" is not the initial object of the word algebra, because the word "zero" is not at all a arity-zero object but part of a very different grammatical system, which may have its own initial object or not. In paranthese I mention that there hat been some linguistics which had been searching for the Ur-words.
5 Deconstructing the origin/Cloning as a mode of iterability
"Il nīy a pas de mot, ni en general de signe, qui ne soit construit par la possibilite de se repeter. Un signe qui ne se repete pas, qui nīest pas deja divise par la repetition dans sa premiere fois nīest pas un signe." Derrida
Similar ideas about the repeatability of signs can be found in the semiotics of Peirce.
The beginning, the start, zero, is not repeatable.
It is not allowed to repeat the beginning.
The whole drama starts with the non-repeatability of the initial object.
This lack of repeatability of zero on one side and the characterisation of signs as repeatable objects on the other side shows some weakness in the argumantation of Rotman. This is the beginning of the whole story of thinking, in mathematics as well in philosophy.
"What would a mark be that one could not cite? And whose origin could not be lost on the way?" Derrida, Margins, p. 321
Derrida brought together in his concept of iterability, not only the stream of non-founded events- the mis en abym, but also the alterity of the iter", the alter". Some Derrida experts seems to be lost in the Strudel des Denkens" (Heidegger) by the infinity of iterability and are blind for this little jump of contexture as hinted by And whose origin could not be lost on the way" which at least is in conflict with the idea of repeatability and which is not necessarily well understand as the far out of sight" by a interminable network".
Is it helpful to mention iteration alters, something new takes place" (Derrida, LI, p.175)? in: Gasche, Mirror, p. 215
Citation goes together, like translation, with jumps to different contextures, all organized by their own and different rules and different origins. More technically, this procedure of citation is possible only in connection with the interplay of an identity and a neighbor function of a mark.
Only conservative translations and citations are covered by category theoretical morphisms. That is, translations which are not loosing their origin in the process of translation. On the other hand, translations as jumps may be wild, out of hands, beyond rules, but not arbitrary. They are ruled" by the procedures" of chiastic change.
The problematics are getting more virulent. Behind this generalized idea of iterability and repeatability which is based on a generalized concept of signs and marks, there is something like the strict non-iterability of the non-signs and non-marks, the kenograms.
In misleading words, more Kantian terms, we can state, that the pure possibility of iterability itself is not iterable. Which doesn't mean at all, that this primordial possibility is unique.
Or even harder: In misleading words, more Kantian terms, we can state, that the pure necessity of the possibility of iteration itself is not iterable. Which doesn't mean at all, that this primordial necessity has to be unique.
Whatīs about the non-iterability of morphograms? What could the term non" mean in a field which is not ruled by negations at all? Is there a a-iterability of morphograms?
Can we repeat or cite or mention a morphogram?
May be, morphograms have the cruel characteristics of not being able to change their definition. Moving morphograms in a kenogrammatical system never changes the morphogram as a morphogram; it remains irrestible to change. To move could mean, to change the context of a morphogram, that is, to put it together with other morphograms, etc. First of all, morphograms are not involved in the game of meaning or significance. They don't mean anything.
How could a morphogram be changed if it is exactly defined as a pattern for all changes of itself as a morphogram?
Morphograms are structurally invariant under negation.
Morphograms can be changed into other morphograms. The reflector operation is a very simple operator to change a morphogram into its inverse morphogram. But that is not a case of iterability in the strict sense.
Is there an iter" for morphograms as there is necessarily an iter for signs to be signs?
To cite a morphogram means to give it a name and to use this name. But this is a change of systems, from the morphogrammatic to the linguistic systems. And not a citation of a morphogram in a morphogrammatic system.
A word, a sentence, a text is part of the game of repeatability. Morphograms may be the invariant and non-significant structure or pattern of all this variations.
We are reaching here a domain or field of pre-semiotics or something like the archi-ecriture, archi-trace and its differance, as is was exposed by the early" Derrida.
6 Where is the problem?
Why should I make such a fuss about natural numbers? Everyone knows how to use them, everybody is counting all the time, our computers are counting permanently, even some dogs are counting...
We have learned all that, and we have taught all that. We are all fit to use them.
But do we understand natural numbers? And why should we understand them? Is it not enough to know how to use them? And are our mathematical considerations not good enough to understand the concept of natural numbers?
As I tried to show, our understanding and our formalization of natural numbers is based on a very deep intuition and an insight in the very nature of numbers.
But how could I presuppose that my robot or my extra-terrestrial visitor has the same deep and well-founded intuitions? How could my robot even have any intuitions?
And surely, nobody has ever seriously asked a human child if it really wants to learn all this stuff.
Obviously, if we want, or have to, construct a robot, that is, an artificial system, which is able to use numbers, we have to be able to teach this system from scratch everything which is needed to understand and to use numbers.
To say, numbers are produced by the process of counting, sounds at first quite good. Counting counts numbers. This pragmatical or constructivist opinion is surely helpful and less magical than the platonist one. But things are not as simple as we are taught they are. Who counts? This could open up an endless enquiry. And it happened and happens somewhere else. The question can be formulated much simpler. If the counter is counting, is then the counter not a product of his own counting? Is the counter himself not as well a product of counting as the numbers he counts? Who is first, the counter or the counted?
Why should we terrorize our understanding of the natural numbers by the simple structure of our language? Or grammar?
To overcome all these circularities, Goodstein has introduced another strategy. We should not deal with numbers, but with numerals. Numerals are elements of a formal game, called arithmetics. Numbers are the interpretations of the moves of numerals in a formal system. The formal system is defined by the rules of arithmetics.
But where are they from? We are back at the beginning of our discussion, the stroke calculus.
|
ThinkArt Lab http://www.ThinkArtLab.com sales@ThinkArtLab.com |
| |
|
|
|