Talking About Models: The Inherent Constraints of Mathematics

Abstract

In this article my primary intention is to engage in a discussion on the inherent constraints of models, taken as models of theories, that reaches beyond the epistemological level. Naturally the paper takes into account the ongoing debate between proponents of the syntactic and the semantic view of theories and that between proponents of the various versions of scientific realism, reaching down to the most fundamental, subjective level of discourse. In this approach, while allowing for a limited discussion of physical and positive science models, I am primarily focused on the structure and ontology of mathematical models, in particular Cohen’s forcing models and to a lesser extent Godel’s constructible universe, to the extent that these were designed to answer questions bearing on the scope, the capacity and ultimately the ontology of models themselves (e.g., the question of continuum), therefore influencing in one or the other way the status of models in general. This status, it is argued, is largely defined by the way models subsume a set-theoretical structure whose constraints, reducible to an extra-linguistic level of discourse, may implicitly condition the epistemic status of models as representations of axiomatic theories. In the last section I deal extensively with the inner constraints of theories (or of corresponding models for that matter) as subjectively originated in a less technical philosophically oriented discussion with certain prompts from phenomenology.