In his seminal paper on "Types, Abstraction and Parametric Polymorphism," John Reynolds called for homomorphisms to be generalized from functions to relations. He reasoned that such a generalization would allow type-based "abstraction" (representation independence, information hiding, naturality or parametricity) to be captured in a mathematical theory, while accounting correctly for higher-order types. However, after 30 years of research, we do not yet know fully how to do such a generalization. In this talk, I interpret Reynolds's insights for the broad scope of mathematics and science, indicate what shape a general theory based on the insights might take, and summarize the work carried out so far.
[An article outline of these ideas, joint with Claudio Hermida and Edmund Robinson, may be found at http://www.cs.bham.ac.uk/~udr.]