ROBERT MARTY
Université de Perpignan
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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 relational structure of type n is a pair (X,a ) where X is a set and a is an n-adic relation on X, that is to say a subset of the Cartesian product Xn. DEFINITION 2. More generally, a relational
structure of type DEFINITION 3. A relational structure
(X, a ) of type DEFINITION 4. Let (X, a ) and (Y, ß) be two relational structures of
the same type n. A function G of X into Y
is said to be compatible if (xl ,x2,
. .. , xn) Consensus Reality as 'Initial Structure' Before proceeding with the main development, let us consider the surmised reality which lies behind multiple (possibly different) perceptions of an object, event or situation. All the relational categories S(F) possess an interesting property which again requires two definitions: DEFINITION 5. A source in S(F) on a set X is a family DEFINITION 6. An initial
structure of a source is a structure (i) (ii) for each object PROPOSITION 1. Adamek [10] The relational category S(F) is initially complete, that is to say every source possesses a unique initial structure. In fact, This has the following consequences
for us: We have a subject (the perceiver) with a set of perceptual judgements
relative to an external object (i.e., a network arrangement of bundles of simple
qualities of feeling) formalized by objects (Yi
, bi ) of the relational category 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. |
| BEFOR | SUMMARY | NEXT |
where each 
is a positive integer, is a pair 
where X is a set and 
a family of ni-adic relations.The
type 
is the arity-size of all participating relations in the "family."
The 
subscript allows there to be several different arity-sizes (each n-adic
relation is like an n-ended directed hyperarcin a hypergaph).
-tuple of 
. In addition, the following definition corresponds to the unity of perceptual
fact relative to an identified object of the exterior world (which itself appears
always as an entity), by means of the introduction of a
connectivity condition on the family 
.
is connected if for all pairs of elements 
and 
there exists a subset of the family 
called 
and elements 
of X such that 
and 
belong to an n-tuple of 
and 
belong to an n-tuple of 
and 
to an n-tuple of a k. (This
generalizes the connectivity of a graph to a directed hypergraph.)
a implies that 
(i.e., membership of a node in a relation hyperarc is preserved).
; 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(Qn) where Qn is the
functor defined by F(X) = Xn and
F(G ) = G n. If 
is the coproduct4 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(Q2 + Q3), 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 Q2 + Q3the 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.
where the 
are objects of S(F) and the 
are maps; we usually denote sources as follows 
on X such that:
is a morphism of S(F)
and each function 
such that 
is a morphism for each i, then 
is also a morphism.
is the required initial structure.
Considering
the intersection (supposed not to be empty) X of the Yi and the inclusion
maps 
we obtain a source
on
X. Proposition 1 assures us of the existence of a relational structure (X, a) which is
incorporated in each 
.
We naturally identify (X, a) with the structure of the object for this subject; 
corresponds formally to the " object behind the appearances" of the phenomenologists. Of course
depends on the 
, that is to say the totality of the experience that this subject has of the perceived
object. It is possible to build a new source associated with a set of perceiving
subjects (a community) taking intersections of underlying sets X of each structure 
presumably built by every subject. Proposition 1 permits us again to assert
the existence of a unique relational structure which can be interpreted as the
structure of the object collectively elaborated by the community in question
(see Figure 2). This is the "common knowledge" of the community members
about the object, or the common conception of it.