Axiomatic Methods in Science

Philosophical analysis of axiomatic methods goes back at least to Aristotle. In the large literature of many centuries a great variety of ssues 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. 1. Theories with Standard Formalization A theory with standard formalization is one that is formalized within first-order predicate logic with identity. Theusual 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 predicate