Axiomatic Typlogy of Empirical Theories

Abstract

The author analyses, within a framework of the structuralist conception of science, different types of axioms that may occur in empirical theories, set-theoretically axiomatized ; in the process, he states six theses, offering formal definítions to characterize each type of axiom, except for the last one. The following set-theoretieal 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 non-empty set
(2) S is a non-empty 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 set-theoretical 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 set-theoretical 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 inter-theoretic 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]