A Refinement of de Bruijn's Formal Language of Mathematics

Abstract

We provide a syntax and a derivation system for a formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples. WTT is a refinement of de Bruijn's Mathematical Vernacular (MV) and hence: – WTT is faithful to the mathematician's language yet is formal and avoids ambiguities. – WTT is close to the usual way in which mathematicians express themselves in writing. – WTT has a syntax based on linguistic categories instead of set/type theoretic constructs. More so than MV however, WTT has a precise abstract syntax whose derivation rules resemble those of modern type theory enabling us to establish important desirable properties of WTT such as strong normalisation, decidability of type checking and subject reduction. The derivation system allows one to establish that a book written in WTT is well-formed following the syntax of WTT, and has great resemblance with ordinary mathematics books. WTT (like MV) is weak as regards correctness: the rules of WTT only concern linguistic correctness, its types are purely linguistic so that the formal translation into WTT is satisfactory as a readable, well-organized text. In WTT, logico-mathematical aspects of truth are disregarded. This separates concerns and means that WTT – can be easily understood by either a mathematician, a logician or a computer scientist, and – acts as an intermediary between the language of mathematicians and that of logicians.