Mathematical Induction Made

Several concise formulations of mathematical induction are presented and proven equivalent. The formulations are expressed in variable-free relation algebra and thus are in terms of relations only, without mentioning the related objects. It is shown that the induction principle in this form lends itself very well for use in calculational proofs. As a non-trivial example a proof of a generalisation of Newman's lemma is given. The paper begins with an introduction to relation algebra and is reasonably self-contained. (Some knowledge of lattice theory, in particular the Knaster-Tarski xed point theorem for complete lattices, is assumed.) The style is expository and suggestions for exercises are included. The basic concept underlying many of the calculations is the notion of a Galois connection, and the paper could be seen as an introductory tutorial in the use of this concept. The idea of formal reasoning | by which we mean the manipulation of uninterpreted formulae according to prescribed syntactic rules | seems to split the computing community into two distinct and opposing schools. There are the enthusiasts who fervently advocate its use, arguing its eeectiveness and reliability, and there are the sceptics who dismiss it, arguing that it ignores the creative process in the discovery of new facts or the design of new systems. Whilst ourselves belonging very much to the enthusiasts we are nevertheless of the opinion that, at this point in time, the sceptics can muster much bigger artillery than we enthusiasts. There are two problems. First, the formal methods community is too concerned with the issue of (a posteriori) veriication of software rather than harnessing formal methods to the much harder task of its construction. Second, it is not suuciently recognised that formal methods must combine precision with conciseness. Too often formal systems are large and complex, involving complex rules with large numbers of parameters,