In December 1920, M. Schönfinkel presented in a report to the Mathematical Society in Göttingen a new type of formal logic based on the concept of a generalized
function whose argument is also a function (Schönfinkel 1924). This mathematical
discipline was subsequently termed combinatory logic
by Curry and "
-conversion"
or "
-calculus"
by Church. Combinators can be used in the study of algebra, topology, and category
theory, and have found application in the study of programs in algorithmic languages.
Combinator
See also
Church Numeral, Combinatory Logic, Lambda CalculusExplore with Wolfram|Alpha
References
Barendregt, H. P. The Lambda Calculus. Amsterdam, Netherlands: North-Holland, 1981.Curry, H. B. Foundations of Mathematical Logic. New York: Dover, pp. 118-119, 1977.Curry, H. and Feys, R. Combinatory Logic, Vol. 1. Amsterdam, Netherlands: North-Holland, 1958.Hindley, J. R.; Lercher, B.; Seldin, J. P. Introduction to Combinatory Logic. London: Cambridge University Press, 1972.Hindley, J. R. and Seldin, J. P. Introduction to Combinators and lambda-Calculus. Cambridge, England: Cambridge University Press, 1986.Holmes, M. R. "Systems of Combinatory Logic Related to Quine's 'New Foundations.' " Annals Pure Appl. Logic 53, 103-133, 1991.Quine, W. V. "New Foundations for Mathematical Logic." Amer. Math. Monthly 44, 70-80, 1937.Révész, G. E. Lambda-Calculus, Combinators, and Functional Programming. Cambridge, England: Cambridge University Press, 1988.Seldin, J. P. and Hindley, J. R. (Eds.). To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. New York: Academic Press, 1980.Smullyan, R. To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic. New York: Knopf, 1985.Statman, R. "The Word Problem for Smullyan's Lark Combinator Is Decidable." J. Symb. Comput. 7, 103-112, 1989.Schönfinkel, M. "Über die Bausteine der mathematischen Logik." Math. Ann. 92, 305-316, 1924.Schönfinkel, M. "Sur les éléments de construction de la logique mathématique." Math. Inform. Sci. Humaines, No. 112, 5-26 and 59, 1990. [French translation with commentary.]Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 711-714 and 1121-1123, 2002.Wolfram, S. "Combinators: A Centennial View." Dec. 6, 2020. https://writings.stephenwolfram.com/2020/12/combinators-a-centennial-view/.Wolfram, S. Combinators: A Centennial View. Champaign IL: Wolfram Media, 2021.Referenced on Wolfram|Alpha
CombinatorCite this as:
Weisstein, Eric W. "Combinator." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Combinator.html