Chancellor, Mathematics provides the descriptive and analytical toolkit that makes much of science and engineering possible. In the last century, computer science has emerged as a new scientific and engineering discipline, radically different from the others, and requiring a distinct mathematical foundation.

Today it is my pleasure to present to you Professor Martin Hyland, a distinguished mathematician whose work sheds light on the nature of computation.

Logicians and mathematicians who work on problems concerning computation often trace their intellectual heritage back to Alan Turing. Professor Hyland enjoys the rare distinction of having a direct claim to such heritage: he studied for his doctorate under the supervision of Robin Gandy at Oxford; Gandy himself had been the only research student of Turing. From the beginning, a central concern of Hyland's research was the question of the structure, behaviour and properties of those functions that can be calculated by a computer, and how to understand them as ordinary mathematical objects. Pioneering work in the 1930s in this area by Alonzo Church - incidentally, Church was the supervisor of Alan Turing's PhD - had introduced a sort of programming language for such functions known as the lambda-calculus. Hyland demonstrated very early on that there was a direct correspondence between the programs in this language and certain collections of mathematical functions. This idea, that programs can be precisely analysed by correspondence with mathematical entities, became the central tenet of the field known as semantics of programming languages; a field on which Professor Hyland's work has had a profound influence.

In 1976 Hyland became a fellow of King's College, Cambridge, where he has remained ever since. During this time, his deep mathematical insight has been brought to bear on a range of questions in mathematical logic, in the area of algebra known as category theory, and in theoretical computer science, with the problems of computable functions a recurring theme. Classical mathematics yields an understanding of functions that is too broad for computation: the early work of Church, Turing and others had clearly established that not all functions are computable. In semantics of programming languages, it turned out that even the most basic notion of mathematics, that of a set, needed refinement in order to support the various operations and constructions that were required. A significant branch of Hyland's work showed how new mathematical universes could be constructed where the basic entities, sets and functions, were immediately amenable to such constructions, and all functions were computable. So mathematics and computation were not so far apart after all.

Among Professor Hyland's more recent achievements, one perhaps stands out. Throughout the 1970s and 80s, much work in semantics of computation was driven by the so-called "Full Abstraction Problem for PCF": the question of how to construct a class of mathematical objects whose behaviour corresponded with the programs of a particular language. In the early 1990s Hyland and his collaborator Luke Ong cracked the problem: replacing sets by certain kinds of games, and functions by strategies for those games, yielded a solution. Far from closing off the field, this result inspired a new generation of research in what is now known as game semantics.

Beyond the unquestionable distinction of his research achievements, Professor Hyland is renowned throughout the research community for his great generosity of spirit, his enthusiasm for the field and his humble concern for those who work in it. His regular appearances at research conferences and workshops worldwide are always eagerly anticipated by audiences keen to benefit from the latest idea or gem of wisdom. Through his work, his insight, and his open, approachable nature, Professor Hyland marks himself out as one of the outstanding mathematicians working at the interface with computing.

Chancellor, I present to you Professor Martin Hyland, who is eminently worthy to receive the Degree of Doctor of Science, honoris causa.

Professor Guy McCusker

Orator