Curry-Howard-Lambek is a set of correspondences between logic, programming, and category theory. You may have heard of the slogan proofs-as-programs or propositions-as-types. These refer to the Curry-Howard correspondence between mathematical proofs and programs. Lambek’s name is appended to the Curry-Howard correspondence to represent connections to category theory.
The term Curry-Howard isomorphism is often used but is an overstatement. Logic and programming are not isomorphic, nor is either isomorphic to category theory.
You’ll also hear the term computational trinitarianism. That may be appropriate because logic, programming, and category theory share common attributes without being identical.
The relations between logic, programming, and categories are more than analogies, but less than isomorphisms.
There are formal correspondences between parts of each theory. And aside from these rigorous connections, there is also a heuristic that something interesting in one area is likely to have an interesting counterpart in the other areas. In that sense Curry-Howard-Lambek is similar to the Langlands program relating number theory and geometry, a set of theorems and conjectures as well as a sort of philosophy.
2 thoughts on “Curry-Howard-Lambek correspondence”
With your use of the term “computational trinitarianism” I want to make a joke about whether the correspondences are homoousiomorphisms or homoiousiamorphisms but fear the reference may be too obscure.
Well played, Jonathan.