A Refinement of de Bruijn's Formal Language of Mathematics

  • Authors:

  • Fairouz Kamareddine;Rob Nederpelt

  • Affiliations:

  • Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K./ E-mail: fairouz@macs.hw.ac.uk;Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands/ E-mail: r.p.nederpelt@tue.nl

  • Venue:
  • Journal of Logic, Language and Information
  • Year:
  • 2004

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Abstract

We provide a syntax and a derivation system fora 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 isformal and avoids ambiguities. – WTT is close to the usualway in which mathematicians express themselves in writing.– WTT has a syntaxbased on linguistic categories instead of set/type theoretic constructs.More so than MV however, WTT has a precise abstractsyntax whose derivation rules resemble those of modern typetheory enabling us to establish important desirable properties of WTT such as strong normalisation, decidability of type checking andsubject reduction. The derivation system allows one to establish thata book written in WTT is well-formed following the syntax ofWTT, and has great resemblance with ordinary mathematics books.WTT (like MV) is weak as regardscorrectness: the rules of WTT only concern linguisticcorrectness, its types are purely linguistic sothat the formal translation into WTT is satisfactory as areadable, 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 computerscientist, and– acts as an intermediary between thelanguage of mathematicians and that of logicians.