natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
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constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Under the identifications
the following notions are equivalent:
A proof of a proposition. (In logic.)
A program with output some type. (In type theory and computer science.)
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)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
In the introduction of
the insight is recalled to have surfaced in the 1970s, with an early appearance in print being the monograph
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
Robert Harper, The Holy Trinity (2011) (web)
Dan Frumin, Computational trinitarianism, Feb 2014 (prezi slides)
An exposition with emphasis on linear logic/quantum logic and the relation to physics is in
A discussion in the context of homotopy type theory is in
For further references see at programs as proofs, propositions as types, and relation between category theory and type theory.
Textbooks on the foundations of mathematics and foundations of programming language which connect via the common theme of type theory/categorical logic include the following:
William Lawvere, Robert Rosebrugh, Sets for Mathematics, Cambridge UP 2003 (book homepage, GoogleBooks, pdf)
Robert Harper, Practical Foundations for Programming Languages, Cambridge University Press (2016) (webpage)
Last revised on November 10, 2018 at 08:15:54. See the history of this page for a list of all contributions to it.