3.31 
The machine nature of the semiotic square need not be
set in sharp opposition to a putative body.
Thought, feeling and doing are connected by feedback
loops which a machine model can emmulate.

3.31 
3.32 
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).

3.32 
3.33 
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.

3.33 
3.34 
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.

3.34 
3.35 
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.

3.35 
3.36 
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 nontechnical 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.

3.36 
3.37 
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 onetoone 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.

3.37 
3.38 
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 nonchange
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.

3.38 