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Before developing the studies that led to the formulation of the generative trajectory, Greimas had demonstrated interest in approaches to communication based in theory of information.  In particular, he exploited the insights offered by the problems of developing machine translation ("La linguistique statistique et la linguistique structurale").  Traces of these preoccupations remain.  For example, the strong thesis of Greimassian semiotics, in the words of Herman Parret, holds that all "meaningful structures and constellations [...] display programs and performances transforming states of being" (Paris School Semiotics xi).

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The terminology of programs and performances recalls cybernetics.  It also recalls the work of a British mathematician on Hilbert's third requirement ­ decidability.  Alan Turing while working on the Entscheidungsproblem produced "a model in which the most complex procedures could be built out of the elementary bricks of states and positions, reading and writing." His biographer, Andrew Hodges, continues

Alan had proved that there was no "miraculous machine" that could solve all mathematical problems, but in the process he had discovered something almost equally miraculous, the idea of a universal machine that could take over the work of any machine.  And he argued that anything performed by a human computer could be done by a machine.  So there could be a single machine which, by reading the descriptions of other machines placed on its "tape", could perform the equivalent of human mental activity. [original emphasis] (Engima 109)

Too bad Ricoeur in his assessment of semiotic formalisation neglects mathematical history and the fate of Hilbert's programme.  For even if he would have difficulty agreeing with the possibilities of machine emulation of human faculties, he would have found an interesting fashion of relating the syntatic and semic aspects of the fundamental grammar of the Greimassian generative trajectory.  Hodges explicates Turing's two different arguments about machine configuration:

From the first point of view, it was natural to think of the configuration as the machine's internal state ­­ something to be inferred from its different responses to different stimuli, rather as in behaviourist psychology.  From the second point of view, however, it was natural to think of the configuration as a written instruction, and the table as a list of instructions, telling the machine what to do.  The machine could be thought as obeying one instruction, and then moving to another instruction.  The universal machine could then be pictured as reading and decoding the instructions placed upon the tape.  Alan Turing himself did not stick to his original abstract term "configuration", but later described machines quite freely in terms of "states" and "instructions", according to the interpretation he had in mind. [original emphasis] (Enigma 107)

In Turing's model the moves are simple.  A symbol being scanned can be changed, erased or remain unchanged; the machine can move to observe another segment (square);  the machine can remain in the same configuration or change to some specified configuration.  Like the semiotic square, past moves determine future moves;  a state may also be treated as an instruction.

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Ricoeur might plead ignorance of Turing's work.  However, since his own critique of the Greimassian generative trajectory targets its completeness and consistency, one suspects Ricoeur of capitalizing on echoes with the work of a mathematician who demonstrated the impossibility of Hilbert's formalist programme.  Kurt Gödel tackled the completeness and consistency criteria of Hilbert's programme and proved the incompleteness of the axioms of arithmetic.  Ricoeur repeatedly claims that Greimas's model is incomplete and inconsistent.

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Gödel's proof, however much it may bolster Ricoeur's rhetoric, raises the spectre of the machine.  Gödel showed "how to encode proofs as integers, so that he had a whole theory of arithmetic, encoded within arithmetic." (Hodges 92) From a semiotic perspective Gödel numbers have a very interesting property for

we can take the number apart like a machine, see how it was constructed and what went into it; which is to say we can dissect an expression, a proof, in the same way.  (Nagel and Newman 1690)

Certainly Ricoeur is not inclined to encode the elements of the generative trajectory into the fundamental structure of signification or the square into itself.  He does come close.  He does discuss the square in terms of mathematical formalizations.

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As the notes to his article indicate, he is well aware of comparisons between the semiotic square and a mathematical structure called the Klein group.  In a non-technical discussion of the Klein group, appearing in 1966 in Les Temps modernes, Marc Barbut explains that two representations of the Klein group

constitute interpretations of it in two distinct languages (endowed with semantics), and therefore they allow a faithful translation from one to the other;  the syntax is the same, only the meaning of the words has changed. (Barbut 376)

In this case, syntax acts between two semantics.  However, in Ricoeur's reading of the semiotic square and the generative trajectory, the equivalence of metalanguages is a question of the relation of a semantics to a syntax.  Ricoeur approaches the problem in terms of the investment of a form with content.  Greimas's terminology of levels encourages Ricoeur's discursive collapse of the question of the adequacy of metalanguages into the problem of fitting investments.

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Mathematical translation, like conversion in the Greimassian generative trajectory, is a function of isomorphism and depends upon a requisite degree of abstraction.  Barbut explains: It is these translations that are called isomorphisms:  two groups (what we are saying here about groups may be said of any kind of structure whatsoever) are isomorphs if they are two representations of the same abstract group;  further, one might add:  if they have the same structure.  This means that their elements may be placed in one-to-one correspondence, such that the image in the second group of the combination of any two elements from the first group is the same as the combination of the images of those two elements.  Isomorphism, the word itself, is plain enough:  the form, the "syntax", the "structure" are the same;  the differences lie, not only in the symbols used to write down the elements ­­ this is trivial ­­ but also in the meaning to be given to the elements;  and one may equally well give them, provided one keeps to the rules, whichever of the possible meanings one wishes. [original emphasis] (377)

Abstraction makes possible the synonymity between structure and syntax.

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Abstraction also enables the comparison of discursive formations including those of mathematics and semiotics.  For example, the Klein group and the semiotic square are not isomorphic.  The Klein group is generated by two rules of combination:  transformations are commutative and each transformation is involutive (n7), that is repeating it twice consecutively changes nothing.  The transformation that changes nothing is represented by an operator and results in one interesting difference between the graphic representation of the Klein group and the semiotic square.  The former represents non-change by a loop at each vertex of a square (Barbut 376).  However this operator and its graphic representation are absent from the semiotic square.  Although he stresses differences between Greimas's semiotic square and the Klein group, this point is not raised by Ricoeur since he works from Piaget's cognitive psychology interpretation of the group.  Piaget's like Greimas's square does not graph operators that produce no change.  Insufficiencies of logical formalization may stem from not enough abstraction, rather than from too much as Ricoeur contends.

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