A pluralist approach to the formalisation of mathematics Journal article, 2011

We present a programme of research for pluralist formalisations, that is, formalisations that involve proving results in more than one foundation.A foundation consists of two parts: a logical part, which provides a notion of inference, and a non-logical part, which provides the entities to be reasoned about. An LTT is a formal system composed of two such separate parts. We show how LTTs may be used as the basis for a pluralist formalisation.We show how different foundations may be formalised as LTTs, and also describe a new method for proof reuse. If we know that a translation Φ exists between two logic-enriched type theories (LTTs) S and T, and we have formalised a proof of a theorem α in S, we may wish to make use of the fact that Φ(α) is a theorem of T. We show how this is sometimes possible by writing a proof script MΦ. For any proof script Mα that proves a theorem α in S, if we change Mα so it first imports MΦ, the resulting proof script will still parse, and will be a proof of Φ(α) in T.In this paper, we focus on the logical part of an LTT-framework and show how the above method of proof reuse is done for four cases of Φ: inclusion, the double negation translation, the A-translation and the Russell–Prawitz modality. This work has been carried out using the proof assistant Plastic.

formalisation of mathematics

mathematical pluralism

logical framework

type theory

Author

Zhaohui Luo

Royal Holloway University of London

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

Mathematical Structures in Computer Science

0960-1295 (ISSN) 1469-8072 (eISSN)

Vol. 21 4 913-942

Subject Categories

Algebra and Logic

Computer Science

Information and Communication Technology

Basic sciences

DOI

10.1017/S0960129511000156