Crítica, Revista Hispanoamericana de Filosofía, Volume 17, number 51, diciembre 1985
Tipología axiomática de las teorías empíricas
[Axiomatic Typlogy of Empirical Theories]
C. Ulises Moulines
Universidad de Bielefeld
Abstract: The author analyses, within a framework of the structuralist conception
of science, different types of axioms that may occur in empirical
theories, settheoretically axiomatized ; in the process, he states six
theses, offering formal definítions to characterize each type of axiom,
except for the last one. The following settheoretieal axiomatization
of classical collision mechanies (CM) is used as ilIustration:
CM (x) iff there exists P, S, T, e, v, m, t1, t2 such that:
(0) x= ≺P, S, T, /R, c, v, m≻
(1) P is a finite nonempty set
(2) S is a nonempty set
(3) T = {t1 , t2}
(4) c maps S onto /R3
(5) v maps P X T into S
(6) m is a function from Pinto /R such that for all p ∈ P, m(p) > O
(7) Σ m(p) . c ° v(p, t1,) =
P ∈ P
Σ m(p) . c ° v(p, t2)
p ∈ P
(where '/R' designa the real numbers and '°', the composition of
functions). This axiomatization is completed with a set of informal
statements that offers the intuitive physical interpretation of the basic
concepts ; e.g., "The elements of P are particles" and "m is the mass
of each particle". First thesis: Among the axioms of an empirical
theory at least one typification occurs. Through the "constructive
echelons" of the projection (π), power set (P) and cartesian product
(X) operations, the typifications make explicit and precise the
type of the settheoretical entities that corresponds to some of the basic
concepts: relations and functions. E.g., for m, we have: m ∈ P (π1
≺P, /R≻ X π2 ≺P, /R≻. The typifications have no empirical
meaning at all; they can be stated by means of pure conceptual
analysis.
Second thesis: Among the axioms of an empirical theory at least
one characterization occurs. Obviously, the typifications do not exhaust
all that can be said in settheoretical terms about the basic concepts.
The characterizations supply additional information which contains,
although in a minimal degree, certain factual meaning, as we can see
in the characterization of m: Func (m) & ∀ p ∈ P(m(p) > O), where
'Func' is a term for the general concept of function.
Third thesis: Among the axioms of an empirical theory al least one
sinoptic law occurs. What type of axiom is (7), the momentum conservation
law? Evidenttly, it is neither a typification nor a characterization;
it is just a scientific law. Laws are the kind of essential
statements in a theory (without a law a theory is not a theory); they
contain the maximum amount and most important empirical meaning
of a theory among all the axioms.
It would seem that the three types of axioms listed are sufficient
to describe an axiomatization of an empirical theory. However, there
are other components that contribute to the identification, which are
necessary for the empirical applications of theories.
Fourth thesis: Among the axioms of an empirical theory at least
one interpretation axiom occurs. An interpretation axiom is a statement
that asserts a link between the potential models of a theory and
the (actual) models of another. These links, as intertheoretic relations,
are vital for the empirical applications of theories; without them
we do not know, e.g., to what kind of things the concepts of mass or
speed are applicable. In CM we need interpretation axioms for the
existing links between itself and, e.g., the chassical kinematics, (physical)
euclidean geometry and chronometry.
Fifth thesis: Among the axioms of many empirical theories at least
one constraint occurs. This type of axioms states a relation among
models of the same theory such that it allows to export data of a
model in order to make e.g., a calculation or a measurement for
a further model. In CM we have constraints for the invariation of
both mass and speed and for the extensivity of mass.
Sixth thesis: Among the axioms of many empirical theories at least
one special law occurs. Is (7) the only law of CM? Certainly not.
In some kind of applications of CM, e.g., the conservation of kinematics
energy law is used. However, this law does not hold in all
applications of CM and, thus, it is not a general law hut rather a
special one. The same situation occurs in many empirical theories, in
particular, in the complexes. But due to the open character of the
set of the special laws, they can hardly be captured in a formal
fashion.
[J.L. Rolleri]
Keywords:
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