This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computeraided proof has forced mathematicians to question the notions of proofs and mathematical truth.

It iswidely recognized that there is a body of mathematics which is crucial to the underpinning of computer science, software engineering, and information and communications technology . For the most part, at undergraduate level, such mathematics isgenerally considered to consist of discrete mathematics, including formal logic. At a research level, one expects to find abstract algebra,category theory , topos theory , etc. In this paper we discuss a body of mathematics which is foundational to formal methods per se and used for the modelling of the usual sort ofsy#264 artefacts: monoids and their morphisms. In particular we show how a single notion of distribution leads to better insights into many of the standard models in use. This discussion leadsnaturally to salient remarks on both the need for, and suitable direction to be taken in, a proposed education reform with a particular emphasis on mathematics for Information Technology .

§0 Background material on sets and functions The language of sets and functions is basic in topology and so it may be useful to have a summary of results that students should already know. Most of this material will have been

§0 Background material on sets and functions The language of sets and functions is basic in topology and so it may be useful to have a summary of results that students should already know. Most of this material will have been