Vernacular Mathematics, Discourse Representation, and Arbitrary Objects

This paper discusses some of the issues that have to be addressed for a systematic treatment of vernacular mathematics. In vernacular proofs, variable letters are introduced and used in a way that does not conform to the usual interpretation offered for referring expressions in the formal semantics of natural language (Kamp and Reyle, 1993; Kamp, 1981; Heim, 1982; Groenendijk and Stokhof, 1990; Groenendijk and Stokhof, 1991; Heim, 1982), although such phenomena do appear to arise in everyday language. These phenomena must be addressed if we wish to use computer aided reasoning systems to check the informal correctness proofs of computer programs, for example. Fine has developed the formal machinery for modelling such uses of variable letters (Fine, 1985), but gives no systematic interpretation that allows us to produce a formal proof from a vernacular one mechanically. A tentative proposal is made as to how the representation of context could be enriched to account for these problematic examples.