Applications of morphic palindromes to the study of Gunther's negative language
Abstract
Gunther's concept of technology is based on a 'melting' of number and notion, Begriff und Zahl, in a polycontextural setting. A key to its study is established by the negation-cycles of polycontextural (meontic) logics that are establishing a negative language.
It is proposed that a morphogrammatic understanding of technology is uncovering a level deeper than polycontexturality and is connecting numbers and concepts not just with the will and its praxeological actions but with 'labor' (Arbeit) in the sense of Marx's Grundrisse ("Arbeit als absolute Armut".)
Palindromic cycles are offering a deeper access for the definition of negative languages than the meontic cycles. Morphogrammatics is opening up languages of creativity and labor.
(work in progress, vers. 0.2, Oct. 2013)
NEWS
The aim of this note is to inform the reader to use the Wolfram Computational Document Format (CDF) Reader for a reasonable access to my new texts published on the website Memristics.com.
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Promoting awareness for a not yet classified crime
FULL TEX:http://memristors.memristics.com/Kindergarten%20and%20Differences/Kindergarten%20and%20Differences.pdfhttp://memristors.memristics.com/Kindergarten%20and%20Differences/Kindergarten%20and%20Differences.html
Kindergarten math: numbers and counting
How it starts
What are the aims of a standard western Kindergarten education in such abstract disciplines like math, geometry and counting?
The answer is easy found. Simply check the offers of one of the many educational organizations and supporting industries.
They all guarantee the parents a steep learning curve for their children to learn to master the basics of the adult mind set of math.
There is not a single offer that is taking the capacities of children seriously and offers strategies to develop genuine infant-adequate education.
One of the many succesfull companies is the company “Home Schooling for Kids” which offers “KS3, A-Levels, GCSE & IGCSE Courses From £350".
www.OxfordHomeSchooling.co.uk
What’s on offer on the ‘Sure Start’ market?
"The goal of kindergarten math curriculum is to prepare children for first grade math. Please see below a list of objectives and goals for kindergarten math:
To count by rote at least to 20, but preferably a little beyond.
The concepts of equality, more, and less.
To count backwards from 10 to 0.
To recognize numbers.
To be able to write numbers.
To recognize basic shapes.
To understand up, down, under, near, on the side, etc. (basic directions).
To have a very basic idea of addition and subtraction.
It also helps to expose the student to two-digit numbers.
"Children may also get started with money, time, and measuring, though it is not absolutely necessary to master any of that. The teacher should keep it playful, supply measuring cups, scales, clocks, and coins to have around, and answer questions."
http://www.homeschoolmath.net/teaching/kindergarten.php
It is also important to know that the definition of a rational human being is implying the skills of those math topics added with the ability to draw some logical conclusions, say with modus ponens.
All that is instructed in the social context of a schooling program that is confusing learning and training with education.
Failing such skills of adult cognition excludes the person to be qualified as a rational human being (homo sapiens).
Without surprise there is some resistance to the schooling movement.
"I suppose it is because nearly all children go to school nowadays, and have things arranged for them, that they seem so forlornly unable to produce their own ideas.” Agatha Christie
http://studentliberation.com/quotes_1.html
In this paper, I will not deal with the many approaches of the anti-schooling movements but with the very essentials of conceptual thinking that are accepted by both sides, the schooling and the anti-schooling institutions and movements.
Some background theories for the traditional approach
Principles postulated in the tradition of the Piaget school
The abstraction principle
"The realization of what is counted is reflected in this principle. A child should realize that counting could be applied to heterogeneous items like toys of different kinds, color, or shape and demonstrate skills of counting even actions or sounds! There are indications that many 2 or 3 year olds can count mixed sets of objects.
The order-irrelevance principle
"The child has to learn that the order of enumeration (from left to write or right to left) is irrelevant. Consistent use of this principle does not seem to emerge until 4 or 5 years of age (German and Galistel 1978).
Constructivism approach
"There is strong evidence that the early teaching of standard procedures for arithmetic problem solving “thoroughly distorts in children’s mind the fact that mathematics is primarily reasoning.” (Kamii et al.1993). In order to address the above problem, new mathematics curricula have been introduced, based on the Piaget theory of Constructivism.
"This approach suggests that logico-mathematical knowledge, apart from empirical or social knowledge (Novick 1996), is a kind of knowledge that each child must create from within, in interaction with the environment, rather than acquire it directly (almost “being donated”) from the environment.”
Natalia Marmasse, Aggelos Bletsas, Stefan Marti, Numerical Mechanisms and Children’s Concept of Numbers
http://web.media.mit.edu/~stefanm/society/som_final_natalia_aggelos_stefan.pdf
These principles are subordinated to the binary question: “To what extent is the sense of numbers innate, and to what extent is it learned?” Nature or nuture?
I learned most, not from those who taught me but from those who talked with me. St. Augustine
Again, Gove’s adviser: “Genes make you smart, not teaching. Genetics outweighs teaching, Gove adviser tells his boss.”
http://www.theguardian.com/politics/2013/oct/11/genetics-teaching-gove-adviser
Without doubt, this poor guy has never studied the miserable arithmetics of modern genetics and DNA research.
There are good reasons to see the decision for a dialog with children as neither belonging to the “nature” nor to the “nuture” camp of the ongoing battle. Dialogs are not in-forming children with educational content but are evoking their own yet hidden capacities of thinking and understanding their own world.
It will turn out that exactly the postulated principles, the principle of abstraction and its corresponding principle of order-irrelevance, has no ‘natural’ foundation in the thinking process of a curious not yet educationally manipulated child. Nor are there any genetical conditions that are forcing to a specific kind of thinking numbers and logic.
Even if there would be strong empirical evidence and verification of a close connection between the concept of number and the genetical prepositions of the humn brain it wouldn’t stop the human mind to surpass such a little handicap.
Up to now we haven't detected any human beings that are able to study the moon with a naked eye nor do we know any genetically privileged children flying around the village without a little helicopter. And, certainly, the whole calculations for the scientific thesis wouldn’t have been realized by non-assisted human brains alone.
Order-relevance is constitutive for the understanding of numbers in the sense of the Stirling abstraction.
A principle of concretization, in contrast to the homogenizing principle of abstraction, is essential for an understanding of numbers in a polycontextural sense.
Gunther’s uncountable objects
The situation today wouldn’t be much different as it was for Gotthard Gunther when he asked, around 1908, his elementary school teacher two serious questions:
1. How is it possible that a simple addition of some single mountains (Berge) results into a mountain range (Gebirge)? That is, 5 Berge = 1 Gebirge, and how works that: 5 = 1!?
2. How is it possible to add different kinds of objects, like 1 church + 1 crocodile + 1 tooth pain + 1 thought together? And how would this relate to the example of the mountain rage (Gebirge)?
Would there be such a monster like a ‘mountain-church-crocodile-tooth pain’ range as a single, albeit complex object, like the addition of mountains is producing a mountain range?
The teacher's answer today will be more or less the same as a child got it a century ago.
Abstraction and enumeration (arithmetization). Obviously, an answer that just moves the question to another level of un-answered questions.
In his biographical text, Selbstdarstellung im Spiegel Amerikas, (1974) he writes:
"Die Arithmetik mußte ganz anderes und Wunderbares leisten können, weshalb er an seinen Lehrer die Frage stellte: Wenn das Zusammensein von vielen Bergen ein Gebirge ergab, was ergäbe dann zahlenmäßig das Zusammensein, wenn man eine Kirche zu einem Krokodil addierte und dazu noch seine Mutter und obendrein ein Zahnweh. (Es ergab sich nämlich, daß gerade zu diesem Zeitpunkt seine Mutter an Zahnschmerzen litt.) Das erschien ihm als eine der Arithmetik würdige und hochinteressante Aufgabe.
"Als man ihm mitteilte, daß man die vier angeführten Daten eben nur als verschiedene Sachen zusammenzählen könne, hielt er das zuerst für ein Mißverständnis und bestand darauf, daß er keine Sachen, sondern eben Kirchen, Krokodile usw. addieren wolle. Und was ändere sich am Addieren, wenn man das Krokodil durch einen Löwen ersetze? Daß sich dann nichts ändere, wollte er nicht glauben.
"Später vergaß er das Problem. Er mußte fast 60 Jahre alt werden, bis es für ihn in der biologischen Computer-Theorie in neuer Gestalt wieder auftauchte.”
http://www.vordenker.de/ggphilosophy/gg_selbstdarstellung.pdf
The development of Gunther’s answers to his early questions went through several stages. From the kenogrammatic approach, to the polycontextural understanding and to a concept that is closely related to his theory of negative languages.
How does abstraction work and how are the natural numbers justified for such a counting process of different objects?
Again, we have the luck to ask Philip Wadler from the university of Edinburgh. His answer is ultimative and should stop any such naive questions for ever.
In his lovely text, probably written for his children and some professors of computer science, he makes it crystal clear:
"Whether a visitor comes from another place, another planet, or another plane of being we can be sure that he, she, or it will count just as we do: though their symbols vary, the numbers are universal.
"The history of logic and computing suggests a programming language that is equally natural. The language, called lambda calculus, is in exact correspondence with a formulation of the laws of reason, called natural deduction. Lambda calculus and natural deduction were devised, independently of each other, around 1930, just before the development of the first stored program computer. Yet the correspondence between them was not recognized until decades later, and not published until 1980. Today, languages based on lambda calculus have a few thousand users. Tomorrow, reliable use of the Internet may depend on languages with logical foundations. "
Philip Wadler, As Natural as 0,1,2
Evans and Sutherland Distinguished Lecture, University of Utah, 20 November 2002.
http://homepages.inf.ed.ac.uk/wadler/papers/natural/natural3.pdf
Gunther was aware that his kind of thinking, and his way of understanding numbers, made him an alien.
Not enough, in his late years he started to develop a system of arithmetics that not only answered his early two crucial questions but it also will be enjoyed by alien intelligence.
He sincerely told his baffeled longtime friend Helmut Schelsky that he isn’t anymore a human being, he just looks like one.
Also the discovery of the zigzag movement of numbers in a transclassical number system is amazing it would be a sign of a serious lack of understanding Gunther’s attempts towards a ‘dialectical’ number theory to celebrate this zigzagging against the ‘Gänsemarsch’ of linearly ordered natural numbers as the sole achievements of Gunther’s polycontextural constructions of the relation of ‘number and logos’.
What could we learn from this story?
Some primitive questions are not necessarily an expression of a lack of rationality but more a sign or symptoms of another, still hidden, pattern of thinking and understanding the world.
Instead of destroying it, a teacher should be able to accept this ‘deviant’ way of thinking and be able to set it into a broader framework of different kinds of rationality.
Talking to the child and developing together new experiences could lead to surprising insights, relevant for the teacher and the curriculum too.
Math for young dancers: Gaps and Jumps
Gaps and Jumps
Where in all those mathematical concepts of successor functions, induction steps, recursion cycles and deduction trees are the gaps and jumps that are natural to dancers?
It surely would be crazy if our numerical counting process would have to stop somewhere at an obstacle, or falling into a counting gap or would have to jump out of such a paradoxical situation.
Why to trust in continuity?
Also I never was a dancer I believe that life without gaps and jumps is grey.
Personally, I was never convinced of this principle of homogeneous continuity necessary for induction, deduction and other step-wise developments of reasoning inside a single paradigm.
On the other hand, if we accept this principle of closure, life gets significantly boring and there is no special motivation to go into it.
Didactical jumps
"First Leah made a jump of three along her number line and then a jump of four. Where did she land?
"Next Leah made a secret jump along her number line. Then she made a jump of five and landed on 9.
"How long was her second secret jump?
http://nrich.maths.org/5652
But an intriguing pre-mathematical question arises too: How does the child know on which number line the jump has to land?
The classical supposition that there is one and only one arithmetical number line possible is not self-evident at all.
Why do we not have different number systems? Greens and reds and blacks?
As we know well, our teacher would explain us that all those differently colored number lines represent the same numbers because we can map each number from one color to the corresponding number of the other color. As they say, number systems are isomorphic. In color terms, they are all grey. And paradoxically, grey itself is not considered as a color.
Why should we accept that?
This principle of homogeneous continuity necessary for induction, deduction and other step-wise developments of reasoning inside a single paradigm has never got my enthusiasm.
On the other hand, if we accept this principle of closure, life gets significantly boring and there is no special motivation to go into it.
Also I was never a dancer I believe that life without gaps and jumps is grey.
What do we learn from this not so innocent example of counting with number lines?
There are at least two different kinds of jumps possible: One inside a linear number system, and one between linear number systems.
Morphic Palindromicity as a Measure for Self-Reference
Abstract
The paper “Gödel Games: Cloning Gödel’s Proofs” started a polycontextural dissemination of the “Beautified Gödel Proofs” of Hehner. This paper takes a morphogrammatic turn to such a dissemination in emphasizing the distributive possibilities of the modi of repetition in morphic palindromes. Palindromicity of morphograms is a precision and concretization of the philosophical construct of iterability as it was applied for a distribution of Gödel’s theorems.
This paper gives an idea of an application of the morphospehere(s) approach sketched in the previous publication.
FULL TEXT
http://memristors.memristics.com/Godels%20Palindromes/Godels%20Palindromes.html
http://memristors.memristics.com/Godels%20Palindromes/Godels%20Palindromes.pdf
General motivation
Morphic self-referentiality is palindromic. Producing finite self-quotational palindromes.
Indicational self-referentiality is a re-entry of the form into the form. Producing an infinite re-entry form. (G. Spencer-Brown)
Symbolic self-reference of a sentence is based on its self-quotation caught by its normalization (Smyllyan) or diagonalization (Gödel). Producing infinite iterations and logical contradictions.
The palindromic approach to quotation as repetition, replication or reflection gives a closer connection between the formula and its quotation than a free, i.e. a structurally not mediated connection to the original formula in a repetition of the formula.
"By the norm of an expression we shall mean the expression followed by its own quotation.” (Smullyan)
All depends now on the understanding of the term “own”. What does “own” mean in the relation to its own sentence. Who owns this sentence that has a relationship to its own?
Obviously, the ownership of both, the sentence and its own quotation is a property of the identity logic and semiotics of logocentrism.
Polycontextural modeling
Again, """In a slogan: "Quotes don't know their mates.""
George Boolos, Logic, Logic, and Logic"
The paper “Gödel Games: Cloning Gödel’s Proofs” started a dissemination of the “Beautified Gödel Proofs” of Hehner. The emphasis was on a complex distribution, reflectional and interactional, of the mechanism of “self-quotation”, ruled by the interplay of quotation and interpretation, as the device of the Gödel construction.
Hence, distribution as a modus of repetition was conceived as “iterative” and as “accretive” and realized in a grid of contextures.
As a further concretization of the idea of quotation and dissemination (distribution and mediation), an application of morphogrammatic palindromic constructions shall be risked.
Because of the lack of any information about the internal differences of the disseminated construction of the polycontextural approach, the chances to fill this gap by a ‘palindromic’ interpretation of iteration shall be taken.
A Turing machine TM M applied on its own ‘description’ [M] leads by diagonalization/normalization to the desired self-referential machine M[M].
M : M ⟶ [M ⇄ M]
Gödel’s and Smullyan’s construction are presuming semiotic identity between the ‘active’ definition of the TM M and the ‘passive’, i.e. the quotation [M] of the active TM M. The relationship between M and [M] is hierarchical, and is modeled as a relationship of program (processor) and data, or ‘operator’ and ‘operand'.
Therefore, the texts of M and [M] have to be semiotically (symbolically) identical. The quotation [M] of M is operationally a ‘mirrored’ and ‘replicated’ text of M.
But this corresponds abstractly the definition of a symmetric palindrome. A mirror-image is the inverse or dual of the mirrored original. In this scenario, the original comes first, the mirrored image second and in reverse order, both together are involved into the relationship or process of self-similar and self-referential mappings and interactions.
This still holds if the replication of the original is mirrored as an iteration of the same identitive structure.
Certainly, that’s not the standard logical and linguistic definition of a quotation as Smullyan's example shows directly: John is reading "John is reading“. Here, the quotation is a literal replication of the first occurrence of “John is reading” by the quote “John is reading”.
Trompe-l'œils of Semiospheres
This surface-structural approach says nothing about the morphic structure of the construction: the composition of the first with the replication of the first as the second in the mode of identity.
To replicate in the mode of identity presupposes a decision in favor for identity, i.e. for equality, in contrast to the possibility of equivalence, similarity or bisimilarity - and others.
Folowing the insights of the morphogrammatic “Trompe-l'œils of Semiospheres” of semiotic configurations we get, e.g. for X = [1,2,3,4], the morphogrammatic result, X"X":
- ispalindrome [1,2,3,4,1,2,3,4];
val it = true : bool
And the semiotic result:
- palindrome [1,2,3,4,1,2,3,4];
val it = false : bool
But not all mirrors are offering a symmetric replication of the ‘original’ text. Asymmetric textures with symmetric functionality are the morphogrammatic subversions, enabled and played by morphic palindromes.
Hence, the Gödelian misery (disapointment) of the limitation theorems for Kurt Gödel is inherited by its complicity with the historical concepts of semiotic palindromes and its symmetry.
Historically, Kurt Gödel met Gotthard Gunther a few years to early to get some valuable hints from Gunther to overcome the negativity of his own results. (Charles Parsons)
There was probably a mismatch of interests too. Gunther wanted some technical help for his attempts to formalize his reflectional logic. Gödel assumed to get some hints for conceptual and philosophical inspirations. Both lacked a mediator to help each other.
The proposed dissemination of “Gödel” in the paper “Gödel Games: Cloning Gödel’s Proofs” wasn't yet contemplating about the difference of ‘symmetrical’ and ‘asymmetrical’ formulations of a Gödel sentence, here in the context of a Turing machine TM and its program M.
What happens if the quotation quotes creatively a different inscription that is nevertheless palindromically equivalent to its origin?
What happens if the process of cloning is accretively, and not iteratively repeating its iterated origin?
Thus, the Gödel theorems are based on symbolic palindromes.
Post-Gödelian theorems are forced by morphic palindromes.
Why would we need self-referentiality, and all its derivates, like self-applications, etc.?
This subversion applied on the definition of the Halting problem has now to consider the double character of self-application, the symmetric and the asymmetric, or as it was conceived before, the iterative and the accretive modi of iterability.
Double confusions
Things might be confusing! A first confusion is cleared by Sipser’s advice:
"Don’t be confused by the idea of running a machine on its own description! That is similar to running a program with itself as input, something that does occasionally occur in practice.” (Sipser, 1997, p.165)
Obviously, all philosophical and logical considerations and problems with self-reference are generously bracket out by Sipser’s advice.
The other confusion is harder to disperse. As shown in the paper ”Morphosphere(s)" palindromes might be asymmetric in the framework of morphogrammatics, i.e. in the paradigm of morphospheres. This possibility of a simultaneity of semiotic asymmetry and a morphic symmetry is the new challenge to the theory of computation.
With the palindromic subversion, the idea of disseminating Gödel’s proof over a polycontextural grid gets a radical concretization, and a possibility of a much more direct elaboration.
Hence, ‘measures’ of undecidability are accessible to intriguing concretions, enabled by the complexity of asymmetric palindromes. Asymmetric palindromes functions as asymmetric quotations and iterations or repetitions and replications.
The formula always presupposes that X≡X, i.e., that X and the quoted (repeated) X in the quotation “X” are equal. It doesn't say that the unquoted X and the quoted X, i.e. “X”, are equal. One is on a first-order, the other on a second-order level. One occurs first, the other second, hence, the second is a repetition of the first. But what is repeated is X and X is equal as X at both positions of occurrence.
Therefore, the nice text-book presentations of Gödel’s theorems, the self-applications of programs and their diagonalizations (Cantor, Tarski, Gödel) and normalizations (Smullyan) are appearing as a tiny special case of identity strategies in the general framework of graphematic scriptures.
This sounds trivial but if iteration alters then the alteration needs to be characterized.
"The defined theory "allow us to replace something with its equal" but it is not able to disallow a distributed substitution because equality of terms is defined in the theory only "up to isomorphism". Such theories are identifying the terms "equal" with "same". The polycontextural approach offers a different option to the difference of equality and sameness. Equality in this sense is an intra-contextural term but sameness is a trans-contextural term. Because substitution is generally defined in a theory only up to isomorphism we always have the possibility to interpret the action also in a trans-contextural way. As long as the definitions of the theory are not disallowing this way to use substitution there is always some degree of freedom to interpret the terms in another similar theory."
http://www.thinkartlab.com/pkl/lola/Godel_Games/Godel_Games.htm
To quote something as something else that still is the same:
Symbolic quotation
To quote something as itself, i.e. in the mode of the is-abstraction as: “X as X is X".
Morphic quotation
To quote something in the mode of the morphic as-abstraction as: “X as Y is Z".
Metamorphic quotation
To quote something in the mode of the metamorphic as-abstraction as: “X as Y is U as V”.
Hence, “Self-reference without reference” is the slogan for the fact that morphogrammatic referencing is evoking its reference in the process of referring to its own reference.
If an ‘expression’ should be quoted (by another expression) it necessarily has to be the identically same sentence of the quotation that has to be quoted. And not accidentally another possible sentence.
Identity secures the truthfulness of the relationship between the original and its replication as a quotation.
This is well codified by: ⌈⌈M⌉⌉ = M. Hence, identity is the measure or criterion of the success of the interplay of the original and the replication of the original on a different linguistic and logical level. It guarantees the successful bridging of the different levels.
Morphograms are not sentences, thereafter there is no ownership by logocentric self-reference and logic possible.
In this sense, it makes, without apophantic reference, sense, to refer to what has no reference, the iterability of inscription, technically realized by morphic palindromes.
If morphograms are not linguistic entities or processes like sentences how could a quotation be realized?
A first attempt to tackle this intriguing question might be achieved by the focus on the kind of iterability between the ‘original’ and its ‘iteration’ or ‘replication’.
Independent of cognitive and linguistic levels of thematizations, the statement of the original and the repetition of it on a different level as a quote, the common or deep-structure of it, is the action of iteration. Here, restricted on a kind of a linear order of the first and the second.
With morphogram X = [1,2,2,3] as the stated ‘original’ and “X” = [2,1,1,3] as the quotation of the original ”itself” as “another” [2,1,1,3] with [2,1,1,3] ≠mg [1,2,2,3], the composed morphogram X"X” = [1,2,2,3,2,1,1,3] is introduced as a ’quotational’ composition. But is it a palindrome? The composed morphogram X”X” is a quotation if it is a palindrome.
Also X and “X” of the example are morphogrammatically equivalent, the composition X"X” = [1,2,2,3,2,1,1,3] is not a morphic palindrome. Albeit the fact that the morphogram [2,1,1,3] is a sort of an iteration or even an accretion of the morphogrammatically equivalent morphogram X = [1,2,2,3], the composed morphogram fails the criterion of palindromicity.
In contrast, the composition X"X” with X = [1,2,2,3] and “X” = [3,1,1,2], X"X” = [1,2,2,3,3,1,1,2] is a palindrome, and is therefore qualified as a morphic quotation. The composition might be written as [[1,2,2,3],[3,1,1,2]] to emphasize its two components.
Hence, the new ‘ownership’ of the ‘reference’ to its ‘own’ inscription is in the ownership and control of what is enabling morphic palindromes. Morphic palindromes own the rules of morphic quotations.
The Trompe-l'œils of Semiospheres
Abstract
Surprisingly, there is a simple key to distinguish and to open up morphospheres in contrast to the semiosphere: symmetric versus asymmetric palindromes. Asymmetric palindromes of the morphosphere are paradox and oxymoric in the understanding of the semiosphere.
Only in the context of human madness and its poetic explosions oxymoric palindromes could eventually occur. Paradoxes, logical and linguistic, are restricted to some crazy word games or mathematical constructions but would never leave their logocentric cage to produce something like paradoxical and asymmetric palindromes.
Morphosphere(s) are opened up by oxymoric palindromes. Morphosphere(s) are the field where asymmetric palindromes get a scientific, mathematical and programmable exposure.
Introduction
Palindromes are well known. "Anna" is one, "elle" is one, and the letter "b" is one, too. Palindromes are read symmetrically forwards and backwards. Both readings result in the same word.
All examples are working with palindromes that are, by definition, obviously, symmetric.
It seems to be very strange to postulate asymmetric palindromes.
But who said that we have to stay in the semiotic sphere, the semiosphere of semiotically founded science and literature?
A glance on morphogrammatics uncovers the funny result that the composition of "anna", "b" and "elle" to "annabelle" is a nice example of an asymmetric morphogrammatic palindrome.
It reads 'forwards' and 'backwards' morphogrammatically as the same.
Albeit it is semiotically, i.e.alphabetically an asymmetric word, and therefore not a palindrome at all, it is a palindrome of the morphosphere.
How does it work?
Some hints are given by the excerpts of the paper
"Morphosphere(s): Asymmetric Palindromes as Keys"
published at:
http://memristors.memristics.com/Morphospheres/Asymmetric%20Palindromes.pdf
http://memristors.memristics.com/Morphospheres/Asymmetric%20Palindromes.html
Strategies towards morphosphere(s)
Also I’m not attracted to the concept of spheres, except, maybe, that of Johannes Kepler’s Music of Spheres, I think it would give my work about graphematics some reasonable positioning if it could be understood as a sphere and could then clearly be differentiated from other spheres, especially from the bio-, logo-, noospheres, but also from Yuri Lotman’s semiosphere.
In this setting, graphematics, as it was introduced in the early ‘70s, is the developing science of studying the morphosphere.
Graphematics contains the studies of
polycontexturality, polycontextural logic, arithmetics, semiotics and programming,
kenogrammatics,
morphogrammatics,
on the different levels of the graphematic system of inscription.
As Lotman introduced his project of a culture-theoretic understanding of systems of sign praxis as the new thematization of the semiosphere in distinguishing it from Vernadsky’s biosphere, my introduction of the project of morphosphere(s) follows Lotman’s programmatic text in a complementary and deconstructive move to elucidate some aspects necessary to establish the new sphere of graphematics, the morphosphere. Leaving a sphere for another new one is not done without use/abuse of past concepts and strategies, and a balance between mimicry, plagiarism and tabu breaking creativity.
Surprisingly, or maybe not so surprisingly, the topos of palindromes, that is leading Lotman’s exposition of the semiosphere, appears as a ‘multi-functional’ key for the establishing of the concept, strategy and project of morphosphere(s).
Sometimes, deep insights are extremely simple.
The key of the distinction between Lotman's semiosphere and the proposed morphosphere(s) has this simplicity, albeit there is no guarantee that this simplicity is also related to a deep insight. The simple difference between the deep-structure of the semiosphere and the deep-structure of the morphosphere is established with the difference of symmetric and asymmetric palindromes.
"The proof that mirror symmetry can radically change the functionality of the semiotic mechanism, lies in the palindrome.” (Y. Lotman)
Palindromes are by definition symmetric. This holds for all occurrences of palindromes in the bio- and the semiosphere.
Asymmetric palindromes are oxymorons.
Enantiomorph oppositions are logically and semiotically dual and are leading to tautologies, avoiding the confrontation with paradoxes, parallaxes, and other monstrosities.
Asymmetric palindromes are presenting the deep-structure of the deep-structure of the semiosphere. It unmasks it as a restricted economy of sings in the mode of linearity and binarity; elaborated as enantiomorphism.
The guide to enter the morphosphere(s) are offered therefore by oxymoric palindromes and their asymmetry.
Oxymoric palindromes are hidden to the phenomenological sight. They cannot be seen and brought to evidence. Up to now there is no insight into the existence of asymmetric palindromes in the scientific spheres of semiotics, linguistics, rhetorics and in corresponding attempts in the sciences of the biosphere, especially in microbiology and genetics.
In contrast, semio-linguistic palindromes needs to be seen. Without the visuality, or other perceptionalities, palindromes cannot be established.
"Thus, the mechanism of the Russian palindrome lies in the fact that the word is seen. This then allows it to be read in the reverse order.” (Y. Lotman)
Palindromes in the Chinese writing
It seems that the complicity of the semiospheric palindromes with Western logocentrism has been observed by Lotman. He confronts his results with the non-logical features of Chinese writing.
"A very curious thing occurs: in the Chinese language, where the word hieroglyph seems to hide its morpho-grammatical structure, reading it in the reverse order helps to reveal this hidden construction, displaying the hidden sequential choice of structural elements in a holistic and visible way.”
"From this, V. M. Alekseev drew the methodologically interesting conclusion: that the palindrome represents the best material for studying the grammar of the Chinese language.
The conclusions are clear:
(1) The palindrome represents the best possible means of illustrating the interrelationship of Chinese syllabic words, without resorting to the artificial lecture-theatre style of displacement and unity exercised by students of Chinese syntax, lacking in skill and talent.
(2) The palindrome represent the best Chinese material for the construction of a theory of Chinese (and perhaps not only Chinese) words and simple sentences. (Alekseev 1951: 102)." (Y. Lotman)
The hidden cannot be seen, it has to be elaborated, uncovered and unmasked by the work of calculation.
An oxymoric palindrome is therefore not given to phenomenological and semiospheric evidence of cognition.
There is nothing to be seen and to be read backwards then.
Paradoxes
Kalevi Kull hints with his paper “Semiosphere and a dual ecology: Paradoxes of communication “ to the importance of paradoxes for the introduction of Lotman’s concept of semiosphere.
"In several of his lectures, Juri Lotman liked to begin his talk with a paradox. Since semiosphere is a very general notion, a description of it via paradoxes might indeed be helpful. A paradox with what it would be appropriate to start here is the famous paradox of learning — Meno’s paradox.”
http://www.ut.ee/SOSE/sss/kull331.pdf
A new way of seeing things has to be learned and trained. This happens with the support of scriptural calculations.
At first it seems to be important to understand that the concept of paradoxes and antinomies is a limited construction depending on the Greek concept and understanding of logos and anti-logos. The study of paradoxes of all kind is having its sophisticated endeavour in the semiosphere.
The structure of perception and cognition in the semiosphere is fundamentally phenomenological.
Despite the dialogical, holistic and intertextual attempts, the modi of perception and evidence in the semiosphere have their foundation in the egology of logocentrism.
Semiospheric studies might be deep-structural studies in contrast to semiotic studies but they remain blind to their own deep-structural decisions.
The semiosphere is not touched by grammatological and graphematical considerations and interventions.
Morphogrammatics of paradoxes
In a radical change of interests and strategies, morphospheric studies, if they still can be called studies without falling back into logocentric complicity with its concept of dialogical truth and rationality, are accepting the work of deconstruction of the very basic presumption of Western culture: its semiospheric umbrella, or as other prefer to say, its logo-phonocentric prison.
The symmetric production rule “S ⟹ a|aSa“ is not considering the asymmetric productions that are morphogrammatically accepted as palindromes, like for example the morphogram [1,2,3,4,1,2] with [1,2,3] = [2,1,4],
[1,2,3,4,1,2], [2,1,4,3,2,1], and tnf[2,1,4,3,2,1] = [1,2,3,4,1,2].
Hence, the context-dependence of the morphic production rule is restricted to symmetric productions with restricted context-dependence.
The filter-method is also not producing constructively the set of palindromes but is filtering them out of the produced trito-universe TU of morphograms.
Morphogrammatic palindrome:
fun kref ks = tnf(rev ks);
- fun ispalindrome l = (l = kref l);
val ispalindrome = fn : int list -> bool
- ispalindrome [1,1,2,2];
val it = true : bool
Symbolic palindrome:
fun palindrome l = (l = rev l);
- palindrome [1,1,2,2];
val it = false : bool
Filtered results of length 6 from TU
Tcontexture 6;
List.filter palindrome “Tcontexture 6";
- length it;
val it = 180 : int
Results for the 31 morphogrammatic palindromes of length 6 from [1,1,1,1,1,1] to [1,2,3,4,5,6]:
[1,1,1,1,1,1],[1,1,1,2,2,2],[1,1,2,1,2,2],[1,1,2,2,1,1],[1,1,2,2,3,3],[1,1,2,3,1,1],[1,1,2,3,4,4],
[1,2,1,1,2,1],[1,2,1,1,3,1],[1,2,1,2,1,2],[1,2,1,3,2,3],[1,2,1,3,4,3],[1,2,2,1,1,2],[1,2,2,2,2,1],
[1,2,2,2,2,3],[1,2,2,3,3,1],[1,2,2,3,3,4],[1,2,3,1,2,3],[1,2,3,1,4,3],[1,2,3,2,3,1],[1,2,3,2,3,4],
[1,2,3,3,1,2],[1,2,3,3,2,1],[1,2,3,3,2,4],[1,2,3,3,4,1],[1,2,3,3,4,5],[1,2,3,4,1,2],[1,2,3,4,2,1],
[1,2,3,4,2,5],[1,2,3,4,5,1], [1,2,3,4,5,6].
Palindromes(6,6) = 31
Symmetric palindromes(6,6) = 5
Symmetric morphogrammatic palindromes of length 6 taken out from TU:
val it =
[[1,1,1,1,1,1],[1,1,2,2,1,1],[1,2,1,1,2,1],[1,2,2,2,2,1],[1,2,3,3,2,1]] : int list list
- length it;
val it = 5 : int
Example :
"Annabelle"
asymmetric palindrome [1,2,2,1,3,4,5,5,4]
"anna" : num(anna) = [1,7,7,1]
” b” = [2]
"elle” : num(elle) = [4,5,5,4]
num(annabelle) = [1,7,7,1,2,4,5,5,4]
- tnf[1,7,7,1,2,4,5,5,4];
val it = [1,2,2,1,3,4,5,5,4] : int
list
ispalindrome[1,2,2,1,3,4,5,5,4]?
val it = true : bool
- kref[4,5,5,4,3,1,2,2,1];
val it = [1,2,2,1,3,4,5,5,4] : int list
Again, morphograms are not defined over an alphabet but by differentiations.
The ENstructure is calculating the differentiation of a morphogram, with N=non-equal and E=equal at the subsystem place.
- ENstructure [1,2,2,1];
val it = [[],[(1,2,N)],[(1,3,N),(2,3,E)],[(1,4,E),(2,4,N),(3,4,N)]]
: (int * int * EN) list list
That is, the trito-normal form tnf of the numeric interpretations of the words “anna” and “elle”, num(anna) = [1,7,7,1] and num(elle) = [4,5,5,4] are equivalent:
tnf[1,7,7,1] = [1,2,2,1] = tnf[4,5,5,4].
But localized in the context of the whole word they are different.
Operation on oxymoric palindromes
Quite obviously there are at a first glance not too many operations possible on asymmetric palindromes that are remaining in the domain of palindromes.
The operation of inversion is part of the definition. General permutations are destroying the definition of palindromes.
Is the ‘addition’(concatenation) of two palindromes a palindrome?
As a metaphor:
You might lock your door with to small keys, say [1,2,3] and [1,2,3,1], but you have to unlock your door with a single key that is the morphic palindromic addition of the smaller keys. For example, with kconcat [1,2,3][1,2,3,1]; there are 5 composed keys of length 7 available.
Coalitions
- kconcat [1,2,3][1,2,3];
- length(kconcat [1,2,3][1,2,3]);
val it = 34 : int
Morphograms: 34.
Palindromes: 14
val it =
[[1,2,3,1,2,3],[1,2,3,2,3,1],[1,2,3,3,1,2],[1,2,3,3,2,1],[1,2,3,4,1,2],
[1,2,3,4,2,1],[1,2,3,1,4,3],[1,2,3,3,4,1],[1,2,3,4,5,1],[1,2,3,2,3,4],
[1,2,3,3,2,4],[1,2,3,4,2,5],[1,2,3,3,4,5],[1,2,3,4,5,6]] : int list list
- length it;
val it = 14 : int
Symmetric palindromes: 1
val it = [[1,2,3,3,2,1]] : int list list
Cooperations
- kmul [1,2,3][1,2,3]
- length(kmul [1,2,3][1,2,3]);
val it = 588 : int
Morphograms: 588.
Palindromes “Palin(kmul [1,2,3][1,2,3])": 44
val it =
[[1,2,3,2,3,1,3,1,2],[1,2,3,3,1,2,2,3,1],[1,2,3,2,1,4,3,4,1],
[1,2,3,2,1,4,5,4,1],[1,2,3,2,4,1,3,1,2],[1,2,3,2,4,1,5,1,2],
[1,2,3,4,1,2,3,4,1],[1,2,3,4,1,2,5,4,1],[1,2,3,3,1,4,4,5,1],
[1,2,3,4,3,1,3,5,4],[1,2,3,4,1,5,2,3,1],[1,2,3,4,1,5,3,6,1],
[1,2,3,4,1,5,6,7,1],[1,2,3,4,5,1,2,3,4],[1,2,3,4,5,1,3,6,4],
[1,2,3,4,5,1,6,7,4],[1,2,3,2,3,4,3,4,1],[1,2,3,2,3,4,3,4,5],
[1,2,3,3,4,2,2,3,1],[1,2,3,3,4,2,2,3,5],[1,2,3,4,3,2,3,4,1],
[1,2,3,4,3,2,3,4,5],[1,2,3,2,4,5,3,5,1],[1,2,3,2,4,5,6,5,1],
[1,2,3,2,4,5,3,5,6],[1,2,3,2,4,5,6,5,7],[1,2,3,4,5,2,3,4,1],
[1,2,3,4,5,2,6,4,1],[1,2,3,4,5,2,3,4,6],[1,2,3,4,5,2,6,4,7],
[1,2,3,3,4,5,5,1,2],[1,2,3,3,4,5,5,6,1],[1,2,3,3,4,5,5,6,7],
[1,2,3,4,3,5,3,1,2],[1,2,3,4,3,5,3,6,1],[1,2,3,4,3,5,3,6,7],
[1,2,3,4,5,6,2,3,1],[1,2,3,4,5,6,3,1,2],[1,2,3,4,5,6,7,1,2],
[1,2,3,4,5,6,3,7,1],[1,2,3,4,5,6,7,8,1],[1,2,3,4,5,6,2,3,7],
[1,2,3,4,5,6,3,7,8],[1,2,3,4,5,6,7,8,9]] : int list list
- length it;
val it = 44 : int
Interpretations
It seems not to too surprising that coalitions (additions, concatenations) of palindromes are accepting some closure under the category “palindrome”.
More surprisingly, at least at a first glance, is the fact that the cooperation (multiplication) of palindromes of the same type are realizing palindromes again.
The coalitions and cooperation of asymmetric palindromes are resulting in asymmetric palindromes as well as in symmetric palindrome. Thus the coalition and cooperation of asymmetric palindromes has a common set of symmetric palindromes.
As a rule it appears that the cooperation of symmetric palindromes is producing a symmetric result.
This could open up some insights into the process of cooperations and coalitions in the context of morphic interpretations of phenomena in the realm of the bio- and semiosphere.
Therefore, systems theory (of what ever level or color) that is successfully applied to the bio- and semiosphere stops to have a successful application in the morphosphere.
In other words, the “Anomaliengrammatik" (Alfred Toth) of palindromes is describing anomalies of the first kind, i.e. symmetric anomalies.
Things are getting hopeless if asymmetric anomalies occur at the desk of our scientifically trained controllers.
What’s the result of the journey?
It isn’t possible for a semiotician to decide what kind of object he, she or it is eying. In the eye of a semiotician, the object he/she eyes is unavoidably a “trompe-l'œil”.
The eyed palindrome is not showing its identity, i.e. its way of coming into existence remains hidden.
The palindrome is eyed as a textual object. It is seen through the eyes of a semiotician.
A semiotic palindrome as an object is hiding its way of construction that would show its character as belonging to the semio- or to the morphosphere.
This objectification of textual events is not prohibiting semioticians to us palindromes as strategies, and strategic tools.
What is studied is the cultural product, not its modi of becoming a product (construction, creation, elaboration). Obviously, semioticians will see it differently.
A semiotic palindrome as a product is always both, a semiotic construction on the base of atomistic concatenation and as the result of a retrograde recursive morphogram.
Because all symmetric palindromes are at a first glance, and without involving them into a constructive play, simultaneously members of both spheres, the semio- and the morphosphere, the semiotic view is blind for this constitutive difference.
As much as palindromes function as a key for semiotic studies, it jumps into the eyes, that the whole endeavour of semiotics is the victim of a sophisticated trompe-l'œil.
Hence, what is eyed is not what is seen. Palindromes cannot be seen. Even the seen palindrome might turn out not to be what had been seen.
The phenomena are not given to the eye of a semiotician, they have to be elaborated by methods not in the reach of the eyes.
More at:
http://memristors.memristics.com/Morphospheres/Asymmetric%20Palindromes.pdf
http://memristors.memristics.com/Morphospheres/Asymmetric%20Palindromes.html
