Abstract

The American Postulate Theorists were a group of influential American mathematicians at the beginning of the twentieth century who presented axiomatized mathematical theories. This group includes, among others, E. V. Huntington, O. Veblen, E. H. Moore, H. M. Sheffer, and C. H. Langford. Views of these men concerning the nature of mathematical theories are explained and criticized. As background, traditional notions of an axiomatic science are analyzed, especially as described by Aristotle, Pascal, and Frege; also discussed are aspects of nineteenth century mathematics considered significant for axiomatic method . Postulate Theorists' own views on the nature of mathematical theories, the relation of a theory to its interpretations, implication, completeness , and definitions are then discussed in detail. ;Some specific results of this examination follow. There is no single conception of "Axiomatic Method," but at last three different conceptions. Some axiom sets are considered foundational for knowledge, some provide simply a clerical codification of knowledge, and some are meant to characterize either one or a set of mathematical objects. The use of models in geometry before 1890 reveals that authors had no clear idea of how to use interpretations to prove independence of the parallel postulate. The conflict between substitutional accounts of variables, which consider them place-holders for constants of the language, and objectual accounts, in which each variable ranges over a class, exists already at the beginning of the twentieth century. Huntington discovered the concept of categoricity to provide an adequacy test for characterizing axiom sets. By Veblen's Thesis, i.e., that any sentence has the same truth value in all models of a categorical axiom set, categoricity also provided a test for completeness. It is a mistake to attribute to Gergonne the notion of "implicit definition" of undefined constants by their appearance in an axiom set. Finally, authors at the beginning of the twentieth century confuse definability of a constant in a theory with interpretability of one theory in another through addition of definitions. In works of Veblen, Russell, Lewis, and Langford this confusion is compounded by use of "parametric definitions" incorporating arbitrary names