An Eclectic View of the Irish School of Constructive Mathematics, from [Lucas 1978] to [Mac an Airchinnigh 2001]

In this paper I celebrate the evolution of the Vienna Development Method (VDM) along its Irish branch and attempt to tell the story that Peter Lucas played in it. There are two parts to the paper. In the first part I tell my story of the early day of the origins of the Irish School of the VDM (VDM ♣), beginning with prehistory in 1978 up until the radical decisions of 1995 which led to the Irish School of Constructive Mathematics (M ♣ c). In 1995 the School committed itself to the development of the modelling of (computing) systems in full generality. This was achieved by embracing Category Theory and by exploring a geometry of formal methods using techniques of fiber bundles. From fiber bundles to sheaves was a natural step. Concurrently, the School moved from the algebra of monoids to categories, and from categories to topoi (alt. toposes). The second part of the paper illustrates, with simple examples, how I introduce topos logic into modelling in 2001. 1 Prologue " We stress here, as was stressed in the introduction to this volume, that the metalanguage is to be used, not for solving algorithmic problems (on a computer), but for specifying, in an implementation-independent way, the architecture (or models) of software. Instead of using informal English mixed with technical jargon, we offer you a very-high-level 'programming' language. We do not offer an interpreter or compiler for this metalanguage. And we have absolutely no intention of ever wasting out time trying to mechanize this metalanguage. We wish, as we have done in the past, and as we intend to continue doing in the future, to further develop the notation and to express notions in ways for which no mechanical interpreter system can ever be provided. [emphasis is mine] " [Bjørner and Jones 1978] p. 33. These words had a profound effect on me and on the development of the Irish School of the Vienna Development Method (VDM ♣). I remember being an angry young(er) man and throwing them back verbatim in the faces of their authors, Dines Bjørner and Cliff Jones, in the 'basement' of the Berlaymont headquarters of what was then known as the