In 1910, the first volume of the Principia Mathematica was published by Welsh mathematician and logician Bertrand Arthur William Russell (1872–1970) and English mathematician and philosopher Alfred North Whitehead (1861–1947). This book was an attempt to put mathematics on a logical foundation, developing logic theory as a basis for mathematics. It gave detailed derivations of many major theorems in set theory, examined finite and transfinite arithmetic, and presented elementary measure theory. The two mathematicians published three volumes, but the fourth, on geometry, was never completed.

On their own, both men did a great deal to advance mathematics. Russell discovered the Russell paradox (see below), introduced the theory of types, and popularized first-order predicate calculus. Russell’s logic consisted of two main ideas: that all mathematical truths can be translated into logical truths (or that the vocabulary of mathematics constitutes a proper subset of the vocabulary of logic) and that all mathematical proofs can be recast as logical proofs (or that the theorems of mathematics constitute a proper subset logical theorems).

Whitehead excelled not only in mathematics and logic, but also in the philosophy of science and study of metaphysics. In mathematics, he extended the known range of algebraic procedures, and he was a prolific writer. In philosophy, he criticized the traditional theories for their lack of integrating the direct relationship between matter, space, and time; thus he created a vocabulary of his own design, which he called the “philosophy of organism.”