On the Completeness Interpretation of Representation Theorems

Abstract

Representation theorems, similar to their counterparts, categoricity theorems, establish an isomorphism between certain algebraic systems. However, in contrast to categoricity theorems, they have received considerably less attention in the philosophy of mathematics. The paper attempts to rectify this shortcoming by excavating the philosophical potential of representation theorems through an analysis of one of their most popular interpretations in the mathematical literature, the completeness interpretation. The meaning of this notion of completeness and the mechanism through which representation theorems are supposed to achieve it are still unclear. The paper addresses both issues. First, it proposes a definition of completeness that best suits the informal notion used in the mathematical interpretation of the theorems. Second, it formally details the mechanism responsible for achieving it. In the process, I’ll issue some remarks on the significance and relevance of the formal reconstruction of the completeness interpretation for non-eliminative structuralism. For exegetical as well as evidential reasons, I’ll focus on Cayley’s representation theorem and use it as a case study.