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
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
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.
