Origins of Recursive Function Theory

For over two millenia mathematicians have used particular examples of algorithms for determining the values of functions. The notion of "?-definability" was the first of what are now accepted as equivalent exact mathematical descriptions of the class of the functions for which algorithms exist. This article explains the notion and traces the investigation in 1931-1933 by which the notion was quite unexpectedly so accepted. The Herbrand-Gödel notion of "general recursiveness" in 1934 and the Turing notion of "computability" in 1936 were the second and third equivalent notions. Techniques developed in the study of ?-definability were applied in the analysis of general recursiveness and Turing compatibility.