What is categorical structuralism? Assessing philosophy of logic and mathematics toda

In a recent paper Hellman [2003], we examined to what extent category theory ("CT") provides an autonomous framework for mathematical structuralism. The upshot of that investigation was that, as it stands, while CT provides many valuable insights into mathematical structure-specific structures and structure in general-, it does not sufficiently address certain key questions of logic and ontology that, in our view, any structuralist framework needs to address. On the positive side, however, a theory of large domains was sketched as a way of supplying answers to those key questions, answers intended to be friendly to CT both in demonstrating its autonomy vis-à-vis set theory and in preserving its "arrows only" methods of describing and interrelating structures and the insights that those methods provide. The "large domains", hypothesized as logico-mathematical possibilities, are intended as suitably rich background universes of discourse relative to which both category-and-topos theory and set theory can be developed side by side, without either emerging as "prior to" the other. Although those domains, as described, resemble natural models of set theory (on an iterative conception) or toposes suitably enriched with an equivalent of the Replacement Axiom, they are defined without set-membership as a primitive, and also without 'function' or 'category' or 'functor' as primitives; all that is required is a combination of 'part/whole' and plural quantification (in effect, the resources of monadic second-order logic). This background logic (including suitable comprehension axioms for wholes and "pluralities") suffices; and ontological commitments are limited to claims of the possibility of indefinitely large domains, any one extendable to a more encompassing one, without end. Two interesting responses to this have already emerged on behalf of CT proponents, one by Colin McLarty [2004] and the other by Steve Awodey [2004]. Here we take the opportunity to come to terms with these and to assess their bearing on our original assessment and proposal. We will begin with a brief review of the main critical points of Hellman [2003]; then we will take up the responses of McLarty and Awodey in turn; and finally, we'll try to draw appropriate conclusions.