ROBERT MARTY Université de Perpignan |
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Using a methodological epistomological process, we now
will associate specific mathematicaluniversals with these empirical universals.
We will use elementary Category Theory in order toformalize Peircean phenomenology,
because Category Theory allows us to treat in a principalway the objects of
the world, and, at the same time, the relations that they have with otherobjects.
(For introductions to Category Theory see [10-12]). The Category-Theoretic notationused
in this paper is taken from Adamek's Theory of Mathematical Structures in
which the study of categories of relational structures is developed.
DEFINITION 1. A
DEFINITION 3. A relational structure
(X, a ) of type is DEFINITION 4. Let (X, a ) and (Y, ß) be two relational structures of
the same type Now we begin to use Category Theory considering a functor F of the category " Set" of the sets within itself, and defining
a relational structure of type F as a pair (X, a
) where X is a set and ; if (Y, b ) is another relational structure of
type F, the function G of X into Y is F-compatible
if . The category of these relational structures with these morphisms
will be called "relational category S(F)." Then the relational
category of type n is S(Q) where Q_{n}n is the
functor defined by F(X) = X and
F(G ) = G ^{n}. If is the coproduct^{n}^{4} of
the
functors we
obtain the relational category of which the objects are defined in Definition 3 and the morphisms in Definition
4. For instance, in S(Q_{2} + Q_{3}), a relational
structure on a set X is a set
consisting of a dyadic relation and a triadic relation ; given the two relational structures of type Q_{2} + Q_{3}the compatible
maps
are precisely those which preserve the dyadic relations as well as the triadic
ones, i.e., implies
and
implies
The specific mathematical universal chosen will be the category the exclusive framework of our formalisation: it contains, in fact, with a suitable
set of index 1, all forms of all possible perceptual judgements of all subjects
and also all possible relations that these forms maintain between themselves.
This gives us considerable power, because from the phenomenological point of view developed in the next section, it is the Peircean notion of "skeleton-set" [8, CP 7-426 ] or the Husserlian notion of "eidos" which we algebraicize in that way. That is the reason why we call this initial structure an "eidetic structure." For example, in Figure 3, we have the eidetic structure of the French flag. |

^{4} The symbol C indicates the coproduct
of a series, which here means disjoint union, also represented by `plus' signs. |

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