Cooper grew up in Bognor Regis and attended Chichester High School for Boys, during which time he played scrum-half for the under-15s England rugby team.

[1] Cooper graduated from Jesus College, Oxford, in 1966 and in 1970 received his Ph.D. from the University of Leicester under the supervision of Reuben Goodstein and C.E.M.

Cooper was appointed Lecturer in the School of Mathematics at the University of Leeds in 1969, where he remained for the rest of his career.

He was a leading mover of the return to basic questions of the kind considered by Alan Turing, and of interdisciplinary developments related to computability.

The book Alan Turing: His Work and Impact, edited by Cooper and Jan van Leeuwen, won the Association of American Publishers' R. R. Hawkins Award.