A History of the Theory of Types

This work traces the development of the theory of types from its origins in the early 20th century through its various forms until the mid 1950s. Attention is paid to the reception of this theory after the publication of the second edition of Whitehead and Russell’s 'Principia Mathematica'. I examine the decline of the theory over four decades, focusing on its modification by logicians such as Ramsey, Carnap, Church, Quine, Godel, and Tarski. Finally, the view of the theory of types as a many-sorted first-order theory is discussed. It is shown that a revival of the ramified theory of types in the 1950s coincided with the consideration of cumulative type hierarchies which substantially altered the form of the theory. The theory was altered even more drastically in its change from a many-sorted theory to a one-sorted theory. This final ‘standardization’ made it not much different from first-order Zermelo-Fraenkel set-theory. The theory of types, whose developments are traced in this work, therefore lost its prominence as the foundation for mathematics and logic. This historical analysis will be of interest primarily to those working in the History and Philosophy of Mathematics.