Philosophical implications of the paradigm shift in model theory

17 January 2019
11:00
Abstract

The paradigm shift that swept model theory in the 1970’s really occurred in two stages. During the first stage in the 1950’s and 1960’s the focus switched from the study of properties of logics to properties of theories. In the second stage, Shelah’s decisive step was to move from merely identifying some fruitful properties (e.g. complete, model complete, ℵ1-categorical) that might hold of a theory to a systematic classification of complete first order theories. Model theorists now undertake a systematic search for a finite set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. With this framework one can compare different areas of mathematics by checking where theories formalizing them lie in the classification. Martin Davis wrote, “ Godel showed us that the wild infinite could not really be separated from the tame ¨ mathematical world where most mathematicians may prefer to pitch their tents. ” We will describe some aspects of the paradigm shift and show how it separates the tame world where most mathematicians pitch their tents from the wild world of arithmetic and set theory. The set theoretically absolute classification of theories separates tame theories such as algebraically closed fields, differentially closed fields, and compact complex manifolds (all ω-stable) and real closed fields (o-minimal) which admit admits (at least locally) a dimension theory from the untamed worlds of set theory and arithmetic, while enabling the study of modern number theory. From the standpoint of the philosophy of mathematical practice the focus is changed from justifying the reliability of mathematical results to the clear understanding and organization of mathematical concepts. This shift raises new or sheds light on a number of philosophical issues. For example, how does formal logic play a significant role in mathematics beyond the old metaphor of ‘the analysis of methods of reasoning’? What constitutes a paradigm shift in mathematics?