Developing a research program
Church earned a doctorate at Princeton in 1927 under Oswald Veblen and returned to teach there in 1929 after research visits. His work concentrated on logic, formal systems, and effective calculability. The program investigated mathematical descriptions of reasoning and computation rather than beginning with the engineering of a particular electronic machine.
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Lambda calculus and calculability
During the 1930s, Church developed lambda calculus using function abstraction and application, later presenting it in The Calculi of Lambda-Conversion in 1941. Explicit rules for variables, substitution, and function operations supplied an analyzable model. It subsequently supported work on programming-language semantics and functional programming without requiring calculations to resemble physical machine instructions.
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Undecidability and Church’s thesis
In 1936 Church published unsolvability results and established the undecidability of first-order logic. Church’s thesis associated intuitive effective calculation with formally computable functions, a claim distinct from a theorem internal to one formal system. Turing studied under him, and comparisons among machine-based and other models clarified the common notion of computation.
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Teaching and organizing logical literature
Church taught researchers including Kleene, Rosser, and Turing. His books and long editorial work for the Journal of Symbolic Logic helped organize an expanding literature. He moved to UCLA in 1967 and continued teaching until 1990; a later collected edition brought together writings, correspondence, and previously unpublished materials.