After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.

In this paper, we explore the role of syntactic representations in set theory. We highlight a common inferential scheme in set theory, which we call the Syntactic Representation Inferential Scheme, in which the set theorist infers information about a concept based on the way that concept can be represented syntactically. However, the actual syntactic representation is only indicated, not explicitly provided. We consider this phenomenon in relation to the derivation indicator position that asserts that the ordinary proofs given in mathematical (...) discourse indicate syntactic derivations in a formal logical system. In particular, we note that several of the arguments against the derivation indicator position would seem to imply that set theorists could gain no benefits from the syntactic representations of concepts indicated by their definitions, yet set theorists clearly do. (shrink)

On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.