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.

Combinatory logic started as a programme in the foundation of mathematics and in an historical context at a time when such endeavours attracted the most gifted among the mathematicians. This small volume arose under quite differ ent circumstances, namely within the context of reworking the mathematical foundations of computer science. I have been very lucky in finding gifted students who agreed to work with me and chose, for their Ph. D. theses, subjects that arose from my own attempts 1 to create a coherent mathematical view of these foundations. The result of this collaborative work is presented here in the hope that it does justice to the individual contributor and that the reader has a chance of judging the work as a whole. E. Engeler ETH Zurich, April 1994 lCollected in Chapter III, An Algebraization of Algorithmics, in Algorithmic Properties of Structures, Selected Papers of Erwin Engeler, World Scientific PubJ. Co. , Singapore, 1993, pp. 183-257. I Historical and Philosophical Background Erwin Engeler In the fall of 1928 a young American turned up at the Mathematical Institute of Gottingen, a mecca of mathematicians at the time; he was a young man with a dream and his name was H. B. Curry. He felt that he had the tools in hand with which to solve the problem of foundations of mathematics mice and for all. His was an approach that came to be called "formalist" and embodied that later became known as Combinatory Logic.

Texte du rabat
This text shows that sufficiently rich combinatory algebras can serve as a platform from which to develop the algorithmic aspects of many areas in computer science, mathematics and their applications. It should be of interest to those studying universal algebra, logic and computer algebra.

Contenu
I. Historical and Philosophical Background.- 1. From Protologics to Combinatory Algebras.- 2. A Brief Recapitulation of Combinatory Algebra.- 3. An Algebraization of Universal Algebra.- 4. Objects Reflected in Their Properties.- II. An Algebraization of Universal Algebra.- Axiomatic Extensions.- Aspects of Universal Algebra in Combinatory Logic.- Remarks on an Algebraic Theory of Recursive Degrees.- III. An Algebraization of Algorithmics.- An Algebraization of Hierarchical and Recursive Distributed Processes.- Algebra of Approximate Computation.- IV. Relations to Logical Computer-Algebraic Calculi.- Solving Discontinuous Differential Equations.- Types and Consistency.- References.