Lambda calculus
In mathematical logic and computer science, the lambda calculus (also λ-calculus) is a formal system designed to investigate function definition, function application and recursion. Although it was originally intended as a mathematical formalism and predates the invention of the electronic computer, it can be seen as the world's first formalized programming language. The programming language LISP was created in large part from the λ-calculus blue-print.
The λ-calculus was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s as part of a larger effort to base the foundation of mathematics upon functions rather than sets (in the hopes of avoiding set-theoretic obstacles like Russell's Paradox). The λ-calculus was ultimately unable to avoid these issues, but emerged as a powerful tool in the field of Computability, the study of which problems are solveable in a systematic way (such as by computer.) Perhaps most famously, it was used to give a negative answer to the Halting Problem, one of Hilbert's Ten Problems posed in 1900 as a challenge to the next century of mathematicians.
The lambda calculus can be thought of as an idealized, minimalistic programming language. It is a close cousin of the Turing machine, another minimalist abstraction capable of expressing any algorithm. The difference between the two is that the lambda calculus takes a functional view of algorithms, while the original Turing machine takes an imperative view. That is, a Turing machine maintains 'state' - a 'notebook' of symbols that can change from one instruction to the next. By contrast, the lambda calculus is stateless, it deals exclusively with functions which accept and return data (including other functions), but produce no side effects in 'state' and do not make alterations to incoming data (immutability.) The functional paradigm can be seen in modern languages like Lisp, Scheme and Haskell.
The lambda calculus - and the paradigm of functional programming - is still influential, especially within the artificial intelligence community.