# nLab computational trinitarianism

category theory

## Applications

#### Constructivism, Realizability, Computability

intuitionistic mathematics

# Contents

## Idea

Under the identifications

the following notions are equivalent:

1. A proof of a proposition. (In logic.)

2. A program with output some type. (In type theory and computer science.)

3. A generalized element of an object. (In category theory.)

This is referred to as “computational trinitarianism” in (Harper), where also an exposition is given.

The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.

Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two. If you arrive at an insight that has importance for logic, languages, and categories, then you may feel sure that you have elucidated an essential concept of computation–you have made an enduring scientific discovery. (Harper)

## References

In the introduction of

• Paul-André Melliès, Functorial boxes in string diagrams, Procceding of Computer Science Logic 2006 in Szeged, Hungary. 2006 (pdf)

the insight is recalled to have surfaced in the 1970s, with an early appearance in print being the monograph

• Joachim Lambek, Phil Scott, Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics Vol. 7. Cambridge University Press, 1986.

See also at History of categorical semantics of linear type theory for more on this.

A exposition of the relation between the three concepts is in

An exposition with emphasis on linear logic/quantum logic and the relation to physics is in

For further references see at programs as proofs, propositions as types, and relation between category theory and type theory.

Revised on September 10, 2014 17:57:54 by Urs Schreiber (141.0.9.60)