S
⊂ x {Vi
| i ∈
I} [Mesarovic, 1972]
Where “x” is a Cartesian
product, “I” an index set, and
V is a set of Vi
relations. The concept of system grew before Condillac
linked to epistemological intuitions in natural philosophy, as we can see en
the works of Galileo[3]
and Newton[4] in
relation to astronomical order and the unity of that order. In this sense, we
can trace its origins to the concepts of wholes
and parts as treated by Plato in Parmenides or Aristotle in Metaphysiscs[5].
Nonetheless, following Buck [1956], we could postulate a valid objection to
this basic core definition as an orderly relation of whole and parts, since it is hard to think of
anything which cannot be regarded as such, rendering the concept rather
useless. However, the use of very general concepts is central to the
epistemological strength of a science, as we see in the development of
mathematics (with variable, function, set), physics (matter, transformation, order), or even in everyday language (thing, whole, part, situation).The fact that system
is a peculiarly ample concept does not disqualify its epistemological validity,
although it complicates its definition.
How
can we maintain the generality of the concept and at the same time give a
specific difference to characterize it?
Gaines [1979] found an ingenious solution emphasizing the human
constructive aspect of the concept: A system
is what is distinguished as a system. This well known definition among systems
scientists, defies traditional formal ones by the use of a double
impredicativity. The obvious one is the apparition of the definiendum in the
definiens, but there is also a meta-impredicativity,
for a definition is a determination, and a determination is an action of
distinction.
Definition
|
|
Definiendum
|
Definiens
|
System
|
What is defined as a system
|
System
|
What pertains to a definition of system
|
The
definiens contains not only the definiendum but also the concept of definition rendering the elucidation
apparently meaningless, and yet, in base to some uncritical common knowledge,
to some everyday social communicative praxis, we can form a meaningful
representation, for we understand that there are many objects that could fit into
the concept. In fact, a further reflection upon Gaynes’ characterization shows
the intention of his fuzzy definition: to provide a very general conceptual
category to be used in a wide variety of intellectual scenarios.
In logical terms, a definition is the declaration that a certain
newly-introduced combination of symbols is to mean the same as some other
combination of which the meaning is already known [Whitehead & Russell.
1967,11], therefore, is a syntactical substitution which implies the extension
of a semantical referent from one string of symbols to another. Although definitions
are neither true nor false, and they do not introduce new semantic referents,
the fact that some definitions imply new combinations of old referents may
produce a semantical expansion. If I define p implies q in terms of the truth table of the disjunction ¬p v q, the semantical expansion is
negligible. On the other hand, if a define subset
as a pair formed by a set and an inclusion map under the condition that any two
elements of such set are equal if and only if their inclusion maps are equal,
the combination of concepts of the definiens expand with their synthesis the meanings
of set, map, inclusion and equality of elements. Furthermore, it
gives a procedure to construct a subset: the pairing of (S,i) and the condition
to distinguish one element from another: for all x1 and x2
∈ S, x1
= x2
↔ i(x1)
= i(x2).
We tend to consider the definitions with null semantical expansion mere
analytical substitutions, and think in terms of definitions when the semantical
content is expanded and the definiens carries some extra information. For this
reason, Gaines definition of system
is somehow unsatisfactory: we expect semantical gaining from definitions, not
only lack of impredicativity. Nonetheless, if we examine closer Gaynes’
explanation of his definition we notice that we can reformulate it like this: a
system is whatever we chose to define as a system, or in other words, a system
is the semantical action of establishing certain distinctions among objects,
whether symbolical or sensorial. Of course, it would be absurd to identify any
semantical action with the definition of system,
for it would not give any specific property for the objects upon which we
perform the distinctions. General speech is a semantical action but it does not
necessarily determine per se the objects considered as systems.
Instead of looking for objects that might qualify as systems, maybe we
should ask instead who makes the determinations of a system?, i.e., who
declares a set of distinctions to be a system? With this procedure we would
avoid ontological hypostases, like Luhman’s [1995,12] declaration that the
concept of system refers to something that is in reality a system, or like
Miller’s [1953] definition of system
as a bounded region of space-time which contain parts associated in functional
relationships, both postulating the existence of the concept beyond the
anthropological action of the scientist. It cannot be proven that the
semantical action and the choice of objects (the restrictions within the
distinctions) are isomorphic to some non-human object, but they can be proven
to be human actions. As Klir [1986] has expressed it, with system we are referring to a human action of abstraction
distinguished on an object by an observer, and such determination reflects the
interaction between the observer and the object. This definition is the line of
Kant’s [2000,691]: a system is the unity of the manifold cognitions under one
idea, a unity which has to be understood, under the epistemological framework
of the First Critique, as equivalent
to the unity based on one principle for the interconnections which constitute
the cognitive action.
We could express the Kant-Klir definition in the following terms: a
system is an ordered human cognitive construction of interconnections which
have been determined according to a unifying principle. It is obvious that such
a definition implies that the unifying principle is given by the choices of the
corporate persona[6] of
the scientist or the philosopher. But the unifying
cognitive principle is far from being only conditioned by scientific
deliberate choices related to a particular science, and we should include the
cognitive conditioning of the human psycho-biological architecture as well as
the social programs of investigation which underlie the first generalizations
of science. Such generalizations are prerequisite conditions for the selection
and ordering of material facts [Dewey. 1938] and they imply the choices of
certain objects and relations from a much wider state space. It seems obvious
to me that those choices are conditioned by homeostatic considerations of
specific human groups, or in other words, that the objects and relations that
get science started follow a principle of utility for the survival of a specific
human group. The principle of survival utility, although based on the general
conditions of homeostasis for biological organisms, contains a further
symbolical dimension in human beings: the conditioning imposed by the acritical
ontoepistemologies developed by a group, the Lebenswelt, which shape and modify the biological conditionings.
Think for a moment about the conditionings that religions had imposed upon
biological human forces, or how their acritical valuations shaped the ways of
life of a community and the development of knowledge. In fact, the
sociocultural conditionings expressed in the Lebenswelt may even affect the biological architecture of the
genome, as the case of the adaptation to lactose proves [Damasio, 2012].
The world of the life of a historical community, the Lebenswelt (L) (in Habermas’ [2010]
sense of the concept) is in a close connection with the realm of experience
that today is under the scrutiny of life sciences. In the philosophical milieu,
one is spontaneously drawn to consider such a realm exclusively under the scope
of contemporary science, and therefore, systematically,
but if we want to elucidate the concept of system,
we should proceed more carefully, for the semantical actions which lead us to
distinguish something as a system are conditioned by some automatic psycho-biological
protocols which belong to a different realm, let us call it Unterlebenswelt (U). The acritical
knowledge which constitute the communicative actions of L is the result of an
evolutive process of communication and complexification which started beyond
human grounds, in the communicative actions of U. Since communication is a
social homeostatic tool, the basic semantics of L are predetermined in the
emotional protocols[7]
of U which enable the acritical character of the linguistic actions of L. In
this sense, L and U define a socio-biological space without which the choices
and distinctions which make something to be a system could never be understood.
L-U, considered as symbolic actions of homeostatic valuation, are not only the
conditions for the formalized symbolic constructions of science, but they are as
well the linguistic core of social identity which renders scientific activity
meaningful. Nonetheless, the formalized symbolic constructions of science are
in turn conditions for those very same choices of the socio-biological space of
L-U, in fact, we are talking of a three dimensional space U-L-Ü (calling
Überlebenswelt the formalized linguistic constructions of science) for the
constitution of the unifying cognitive principle from which we construct the
concept of system.
In this sense, an exclusively mathematical characterization of the
concept of system seems insufficient and the definitions carry some serious
ontoepistemological problems that can heavily weigh upon the praxis of systems
science. Let us take the current definition that most system scientists would
approve, and say that S is a general system if it consists of an ordered pair
of sets (M,R), where M is the set of
objects of S and R the set of some relations among those objects [Lin, 2002]. From
a constructivist point of view both M and R are problematic concepts. Any
theorem that uses the axiom of choice (present in the ZFC list of axioms) is
questionable, for such an axiom is equivalent to a principle of omniscience,
like Bishop [2012] correctly remarked. But our list of troubles does not end
here. As it is well known since Russell, relations can be paradoxical:
(x)(y)
(<x,y> ∈
z ↔ Pxy),
for Pxy an open sentence on ‘x’ and ‘y’
(1)
Now
substitute in (1) ‘Pxy’ for ‘<x,y> ∉ z’, then we have Russell’s paradox.
Furthermore, if we are to
define relations as ordered pairs, we would need the axiom of choice to declare
that every set can be well ordered and construct our relation. But even if we
admit the axiom of choice, we can construct systems out of well defined
numerical sets which, although related through the composition of a Cartesian
product, have no computable relation, like the case of the position and the
velocity of a particle in a 𝔖
quantum physical system (due to Heisenberg’s uncertainty principle). Set and relation are not independent concepts. How can we construct a set
without the concepts of a relation of
equality among its elements and a membership
relation? And how can we construct a relation without the previous concept
of set? How are we to consider their
dependence? These problems are not solved either by the construction of the
concepts of set and relation from the calculus of predicate
logic. Modern mathematics, whether Platonist, formalist or constructivist
cannot do without the concepts of set
or relation, but the grounds for
these concepts is not to be found in what we consider the practice of the
mathematical science. In fact, such concepts seem to be developments of basic
intuitions of our cognitive processes as proposed by Dehaene [2011], intuitions
that were complexified through
progressive historical developments of the human thinking, as we see in the
intuitive set theory of the totemic thinking, or in the Babylonian science of
lists [Munoz, 2013].
An
exclusively mathematical definition of the concept of system can only be done at the prize of meaningless constructions, rendered
meaningful only through the praxis of social communication, i.e., when they are
subsumed under a Lebenswelt
interpretation of the concept of system,
but not from the grounds which purportedly sustain it formally. Since systems
are linguistic objects, they are conditioned by Tarki’s [1983] semantical
theorem, therefore, the semantical image of a system, i.e., its concepts of meaning, definition, truth, and
the like, cannot be produced exclusively with the elements of that system.
According to the theorem, to define a formalized linguistic system we need two
linguistic morpho-syntactical systems of different order, the object language
and the metalanguage. However, it is interesting to notice that any composition
of semantical concepts in the metalanguage asks for a further elucidation in an
extra-meta-language, as we see with the aforementioned restriction of the concept
of system to Tarski’s theorem, for the theorem, which gives the conditions of a
semantical distinction (true-false, etc.) requires the concept of system, but system requires in turn the concept of semantical distinction. And something similar happens with the concept
of definition which we use as a tool
to represent the concept of system,
for the concept of definition, as
used in logic and formal reasoning, i.e. in Ü, is already a system with two
objects (definiendum and definiens) and an isomorphism.
Any
formalized definition or restriction of the concept of general system meets the syntactico-semantical incongruence of a
Gödelian language (any language with the axioms of Principia Mathematica plus
the Peano axioms). Semantic actions can be formalized bounding the universe of
discourse with a limit or limen which is never absolute and never grounded on Ü.
The objects of a general system are symbols, and therefore, objects under an
interpretation, particularly the valuation performed by a specific
philosophical or scientific historical community which lives under the
constrains of a particular U-L-Ü symbolic configuration. For that reason,
Klir’s [1991 ]concept of general system
as an interpretation-free system chosen to represent a particular equivalence
class of isomorphic systems, has to be considered in the context of the
practice of systems science but not from a theoretical point of view. The
different systems of the sciences, physical, biological, social, etc., are some
sort of models of the concept of general
system, but this latter concept is not interpretation-free, but bounded to
epistemological restrictions (social and psycho-biological). Particular systems
are under two types of interpretation: one is their modelization of M and R of
the system, and the other is the U-L-Ü interpretation which share with general
systems. In order to avoid the theoretical entanglement of the foundations of
mathematics, we should need an intuitive framework for the characterization of
the concept of system.
The
wide scope of systems science does not necessarily have to follow the only way
of mathematics. Furthermore, it could not accomplish its goals based only on a
mathematical approach. The concern for a thoughtless misuse of mathematics in
scientific practice (especially among social sciences) was already expressed by
Bishop [1975] who asked for a more meaningful application of the mathematical
conceptual tools. The epistemological objections presented in this paper to the
current definitions of system do not intend to reject mathematical thinking
(that would be impossible, for the basic intuitions of mathematics are
biologically conditioned), but simply its implementation with judgment, i.e., with
self-criticism and meta-theoretical tools. I do not think that the idea of an exact philosophy, as Bunge [1977]
pretends for system science, could be sustained outside a naive Platonist ontoepistemology,
in fact the computational character of the praxis of systems science seem to
favor finite and constructive processes. But I think that systems science, if
it wants to become the science of
sciences, or put in a less Biblical fashion, the ground for contemporary epistemology,
could benefit from a philosophical thinking oriented to the questions of
meaning in the human symbolic constructions. The proposal of a U-L-Ü framework
as a conceptual referent for the unification of cognitive principles which may
help to the understanding of the concept of general
system is grounded in the different types of human linguistic actions,
starting from the emotional protocols shared with mammals, which give the
semantic bases for human social homeostasis, and ranging all the way to the
most complex symbolical human constructions.
Reference
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[1] A system is nothing but the
disposition of the different parts of an art or a science in such an order that
the parts sustain reciprocally each other, and the last ones are explained by
the first ones. [Condillac, 1798].
[2] For Ackoff a system is a set
of interrelated elements with 3 properties: each part has an effect on the set
as a whole, each part depends at least on another part and cannot have an
independent effect on the whole, and finally, every possible subgroup of elements
has the two previous properties. [Ackoff, 1973].
[6]
I have treated the concept of corporate
persona somewhere else. [Munoz, 2013]
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