Axiomatic Methods in Science

Abstract

Philosophical analysis of axiomatic methods goes back at least to Aristotle. In the large literature of many centuries a great variety of issues have been raised by those holding viewpoints that range from that of Proclus to that of Hilbert. Here I try to consider in detail only a highly selected set of ideas, but they are ones I judge important.The first section gives a brief overview of the formalization of theories within frrst-order logic. The second section develops the axiomatic characterization of scientific theories as set-theoretical predicates. This approach to the foundations of theories is then related to the older history of the axiomatic method in the following section. Theories with Standard FormalizationA theory with standard formalization is one that is formalized within first-order predicate logic with identity. The usual logical apparatus of first-order logic is assumed, mainly variables ranging over the same set of elements and logical constants, in particular, the sentential connectives and the universal and existential quantifiers, as well as the identity symbol. Three kinds of nonlogical constants occuf, the predicates or relation symbols, the operation symbols and the individual constants.The expressions of the theory, i.e., finte sequences of symbols of the language of the theory, are divided into terms and formulas. Recursive defrntions of each are ordinarily given, and in a moment we shall consider several simple examples. The simplest terms are variables or individual constants.New terms are built up by combining simpler terms with operation symbols in the appropriate fashion. Atomic formulas consist of a single predicate and the appropriate number of terms. Molecular formulas, i.e., compound formulas, are built from atomic formulas by means of sentential connectives and quantifiers. A general characterization of how all this is done is quite familiar from any discussion of such matters in books on logic. We shall not enter into details here. The intuitive ídea can probably be best conveyed by considering some simple examples. In considering these examples, we shall not say much of an explicit nature about first-order logic itself, but assume some familiarity with it. In a theory with standard formalization we must first begin with a recursive definition of terms and formulas based on the primitive nonlogical constants of the theory. Secondly, we must say what formulas of the theory are taken as axioms.' (Appropriate logical rules of inference and logical axioms are assumed as available without finther discussian.) 205M. E. Carvallo (ed.), Nature, Cognition and System II. 205-232.

Foundations of applied mathematics I

“…There is no simple or elegant way to include this mathematical substructure in a standard formalization that assumes only the apparatus of elementary logic. (Suppes (1992): 207)…”

What’s Right with a Syntactic Approach to Theories and Models?

“…] Like Van Fraassen, Suppe (1989) and others also suggest describing a theory by identifying a class of structures, and in the following, I will focus on such semantic approaches. When relevant, I will note the relation of the discussion to the semantic approaches of Suppes (1992) and Stegmüller (1979), for example, which rely on set-theoretic formalizations. Semantic approaches are widely thought to avoid a number of perceived shortcomings of syntactic approaches: (i) Syntactic approaches often have unintended models, unlike semantic approaches, (ii) syntactic approaches require an account of the relation between language and the world, (iii) semantic approaches are language independent, (iv) the relation between theory and observation is misleading or wrong in syntactic approaches, (v) the description of scientific theories in syntactic approaches is cumbersome, (vi) with their focus on models, semantic approaches are closer to actual scientific practice.…”

Reflections on the revolution at Stanford

“…Beth also adds the requirement that the meaning of the concepts in Π c must be clear so as to not require any further explication. 22 Suppes [1992] has pointed out that Aristotle did not provide a single example of a piece of science that qualifies as achieving the classical ideal by his own lights (ibid., p. 215); the very first piece of science that qualified was Euclid's Elements. We would classify Euclidean geometry today as a piece of pure mathematics.…”