Semiotic Constraints of the Biological Organization 213description of (M,R) systems, and the semiotic aspect of this theory is also evident (Kull,
1998).
In Rosen‘s concept, the (M,R) system is viewed as a conjunction of three mappings F, Φ,
β uniting two sets A and B that have correspondingly elements a and b:
In this system of mappings, the following assumptions are set:^
F(a) = b,
Φ(b) = F,
β(F) = Φ with β equivalent to b.
We can write Φ ⊢ F ⊢ b ⊢ Φ, where the symbol ⊢ means ―entails‖ (Kercel, 2007). The
map does not merely entail the result but contains it, which becomes a consequence of the
semiotic relation when we have a closed significative structure of a bootstrapping type.
A formal entity β has the property that β(F)=Φ. Thus, β is a mapping between Map (A,B)
(the set of possible metabolisms) and Map (B(Map(A,B))) (the set of possible selectors). The
procedure defined by β consists in the operation that, given a metabolism F, produces the
corresponding selector Φ that selects metabolism. For β to exist it is required that the equation
Φ(b)=F , for Φ must have one and only one solution. The most fundamental question here is
if it is possible to produce a definitive mathematical description (preferably algorithm) to
calculate β. Rosen recognized that the existence of such mathematical description is
mathematically difficult (Rosen, 2000, pp. 261–265). The tricky thing here is that β is a
function, but it is simultaneously ̳ ̳equivalent‘‘ to an element b B, in the sense that β sends
any f to the unique Φ such that Φ(b)=F. As a result, this construction solves the problem of
infinite regress. The infinity generates its limit within the system by allotting certain element
b with the property reflecting the whole system. Thus the element b gets a dual function
becoming a sign (argument) designating the system as a whole. It has a semiotic nature
imposing the semantic closure to the system in which it is present.
The main problem that we face in formalization of (M,R) system is that in general we do
not know for certain which quantifiers to put on the elements b and a. The original definitions
of the (M,R) systems give no information about the element a in A and b in B. In other words,
the question is if we have for some b in B and some a in A such that b=F(a), there is a unique
Φ in Map(B(Map(A,B))) such that Φ(b)=F. The assumption is affirmative, but it is not well
defined and proven in the theory and still remains unclear. The solution can need some fuzzy
and soft mathematical approach that would introduce the functional elements of (M,R)
systems as results of a kind of a selection procedure taking place in the potential field. This
selection procedure will provide a minimum price of action (Liberman, 1989) for the
maintenance of a given organizational invariance. Here we face the intervention of
mathematically undefined parameters linking elements, maps, and functions in the same way
as the physical fundamental constants entail physical universe so that it becomes structurally
stable and, in principle, observable. Like in Feynman‘s quantum mechanics, the real path
becomes a result of integration of all possible paths, the selectors Φ and β in Rosen‘s (M,R)
systems emerge through a kind of a selective evaluation procedure taking place on
metabolisms and replacements. These selections are semiotically arbitrary and can be derived
only through the evaluation of optimality of a possible selector solution.
A B Map A B Map B Map A B F
, ,
β^