- The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another which makes it difficult to know where to get on and where to get off the circle of explanation. As I currently understand their relationship, it can be approached in the following order.
- Relations have types.
- Types are functions.
- Functions are relations.
In this setting, a type is a function from the places of a relation, that is, from the index set of its components, to a collection of sets known as the domains of the relation.
My 3-basket mantra recited above harks back to the mid 1980s when I took a course on Applications of Lambda Calculus from John Gray at Illinois. It was all about categories, combinators, and computation, focusing especially on Cartesian Closed Categories, one of the hot topics of the day. We had a packet of readings from the classic sources and used J. Lambek and P.J. Scott’s Introduction to Higher Order Categorical Logic as our main text. I followed that up with a supervised independent study where I explored various themes of my own.
The directions I pursued and continue to explore all have to do with mutating category theory just far enough to encompass Peirce’s 3-eyed vision in a more natural fashion.
I’ll make that more explicit when I next get a chance.