Concept
Curry–Howard correspondence
Overview
In programming language theory and proof theory, the Curry–Howard correspondence is a direct relationship between computer programs and mathematical proofs. It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation.
It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and the logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Curry–Howard correspondence is.
The extension, figure, divisibility, and mobility of Colour, the sole object of Sight, though, on account of their correspondence and connection with the extension, figure, divisibility, and mobility of Solidity, they are called by the same name, yet seem to bear no sort of resemblance to their namesakes. As Colour and Solidity bear no sort of resemblance to one another, so neither can their respective…
Adam Smith, The Essays of Adam Smith
States what Curry–Howard correspondence is.
To complete this rapid general enumeration of the different essential parts of the calculus of direct functions, I must, lastly, mention expressly one of the most fruitful and important theories of algebra proper, that relating to the transformation of functions into series by the aid of what is called the _Method of indeterminate Coefficients_. This method, so eminently analytical, and which must be regarded as one…
Auguste Comte, The philosophy of mathematics
States what Curry–Howard correspondence is.
Philosophers who discussed Curry–Howard correspondence
Related ideas
- Analogy
- Springer Science+Business Media
- Free variables and bound variables
- Kurt Gödel
- Intuitionistic logic
- L. E. J. Brouwer
- Logic programming
- Logical conjunction
- Proof theory
- Modus ponens
- Mathematician
- Intuitionism
- Hypothesis
- Sequent calculus
- Academic Press
- Set theory
- String theory
- Stephen Cole Kleene
- Modal logic
- Intuitionistic type theory
- Logical consequence
- Modal operator
- Formal system
- Type theory
Place in the tradition
Part of Intuitionistic logic.
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