Origins of recursive function theory

For over two millennia mathematicians have used particular examples of algorithms for determining the values of functions. The notion of "λ-definability" was the first of what are now accepted as equivalent exact mathematical descriptions of the class of all number-theoretic functions for which algorithms exist. This article explains the notion, and traces the investigation in 1931-3 by which quite unexpectedly it was so recognized. The Herbrand-Gödel notion of "general recursiveness" 1934, and the Turing notion of "computability" 1936 were the second and third of the equivalent notions. Techniques developed in the study of λ-definability were applied in the analysis of general recursiveness and Turing computability.