From a mathematical point of view, it is impossible for the recursive calculus(reisonnement par rcurrence) to have an additional relationship/property, i.e. in addition to "part to whole" also to have "part to part" ?
Example is somewhat confused. It is not certain whether part-to-whole relation, namely parthood, should be distinguished from part-to-part relation. For the latter relation is just a parthood where the ‘whole’ side is a part of some other entity, at the same time.
To make the distinction required, you may define part-to-whole relation as a type-specific property, namely parthood*. Parthood* would be defined on the set product P×W, where W is any ‘proper’ wholes and P is any parts.
Then parthood* would not be a recursive property anymore, as you suggested. But this is not because recursive properties cannot be defined from the mathematical point of view, but we made our definition in a type-sensitive way.
My question arose when I read in the book Epistemology of Logic by the Greek logician Epameinondas Xenopoulos page 285 (8.1) The dialectical character of contradiction
I am giving you a part of the above section, but I don't know if my translation by the help of google can give exactly the correct meaning:
“ 8.1 The dialectical character of contradiction
As we have formulated above, logic in general, in its historical course, has the "tendency" to "overcome", every time, its judgment in a dialectical way. The same happens, as G. Bachelard (*C. Bachelard, Epistemologie, PUF, 106). cautions us, in the area of science and knowledge in general, showing us the dialectical character of their development. For example: Euclidean geometry was opposed by non-Euclidean’s, Archimedes' lines by non-Archimedean lines in Newton's physics, Einstein's physics( F. Couseth, Les fondements des Mathematiques, 1936, σ. 38-39) etc. We see, a posteriori, the dialectical way of developing them together and of their methodology. Their "main" characteristic is the development of contradiction, which consists, at the real level, of: "if we posit T as a position, then we deny it in the form of a non-T opposition and then we "overcome" both ( position and opposition) and we keep the essential of T and not-T, with the emergence of new properties, lifting the above contradiction into a: "reconciling-synthesis" S, that is: T. -T = S. We can, here, observe that the composition S encloses three ideas, as J. Piaget( J. Piaget, Les problemes principaux de l' epistemologie mathematique, στο«LCS» (σ. 554-594), σ. 594) writes, namely: negation, "overcoming" and the preservation of "opposite" elements within the bosom of overcoming. Fundamentally, the "motive" of overcoming is the "products of the S synthesis", within which the "overturning" of the meaning of the system takes place. This reversal takes place by virtue of negation or reversal (inversion), recurrence, reciprocity and correlativité and in general the INRC operators, which are the products of the composition S and definitely constitute the basis of retrospective reasoning (J. Piaget, logique, σ. 370).Recursive reasoning (raisonnement par recurrence): is the passage of the element towards the whole with the "gradual composition" of the parts that are in function with each other. Basic type of recursive reasoning is:[ (P0) . (Pν ⊃ Pν+1) ] →Px which means: "if P is valid for 0" and "P is valid for ν" then "P is valid for ν+1" implies: "P is valid for x (or 0…..∞ )".Thus the implication: (Pv → Pv+1), in contrast to the two-valued implication (p→q), expresses a "law" of construction and not a "simple encapsulation" (enclosure) of "part to whole" only, but also a relation " part to part". In binary logic the construction is done by simple generalization. Conversely, in dialectical logic or recursive calculus, we have a construction based on "extended induction". The contrast is obvious: logico-mathematical generalization is contrasted with extended induction, which is due to an extension of the dialectical process. Because not-T can be: subversion, reciprocity, variability, negation, etc., where, here, we also have a reversal of meaning (overturning the concept of meaning ), and this precisely constitutes the cause of overcoming with dialectical diversity "which does not it is limited only to "yes" and "no", writes Piaget(J. Piaget, ο.π., σ. 376.), but also extends to "the same and the other" (a position already expressed by Plato in the Sophist) and all this according to all its levels: " parity and non-parity".”
The contrast is obvious: logico-mathematical generalization is contrasted with extended induction, which is due to an extension of the dialectical process. Because not-T can be: subversion, reciprocity, variability, negation, etc., where, here, we also have a reversal of meaning (overturning the concept of meaning ), and this precisely constitutes the cause of overcoming with dialectical diversity "which does not it is limited only to "yes" and "no", writes Piaget(J. Piaget, ο.π., σ. 376.), but also extends to "the same and the other" (a position already expressed by Plato in the Sophist) and all this according to all its levels: " parity and non-parity".”