The Combinatory Programme

The programme in combinatory logic, developed at the ETH Zürich, takes its philosophical basis primarily from the work of two mathematicians of widely seperated eras. The first was Richard Dedekind who was probably the first to base the development of mathematics on pure thought. The second was Haskell Curry who, as a young man, took in the task of creating a formal basis for the foundation of all mathematics. Probably neither of them would have foreseen the extension of their ideas to a profound influence on a discipline that did not even exist in their time. For the purpose of the programme is no less than to rework the mathematical foundations of computer science on such a theory of pure thought. It begins from the idea that, if logic is to be the science of correclty dealing with thought-objects, the underlying theory must be in some sense a part of, or at least a preliminary to, its structure; i.e., a protologic. From this idea a combinatory algebra is constructed, using a programmatic mixture of the classical axiomatic and set-theoretic approaches.§This monograph is the result of a sustained of effort on the part of the author and the group of the students who agreed to work with him to put combinatory algebra in the center of the foundational structure of computer science and related mathematics. It shows that sufficiently rich combinatory algebras can indeed serve as a platfrom from which to develop the algorithmic aspects of many areas of computer science, mathematics and their applications. The book will provide a rich experience for scolars and students interested in such topics as universal alebra, logic, computer algebra, recursion theory, and many other topics that stand at the heart of theoretical computer sciences.