Abstract

Two remarkable results attained by Domain Theory, which serves as mathematical foundations for denotational semantics of programming languages, are explained and considered from philosophical viewpoints: 1) the analysis of recursion by the fix-point semantics and 2) the introduction of the notion of continuity and of compact elements. In particular, the author finds them conceptually illuminating in that firstly, they succeed in making explicit those unnoticed semantic elements lying behind the syntax of the languages which play essential roles in the construction and execution of recursive programs, and that secondly, they show the way to reconstruct various ordinary classical mathematical structures by virtue of complementing approximation processes to their infinite noncompact elements.