In 1879 Gottlob Frege published the Begriffsschrift, founding modern mathematical logic, and from 1893 onward he attempted in the Grundgesetze der Arithmetik to derive all of arithmetic from pure logic. That programme — logicism — held that mathematics is not a special kind of knowledge but a set of consequences of logic itself. Bertrand Russell, a mathematics student at Trinity College, took up the project around 1900 and from 1903 began working on it jointly with his former teacher Alfred North Whitehead. Writing and revising took a decade, the sustaining force of inherited wealth, and a daring publishing decision by Cambridge University Press.

The obstacle that nearly broke the programme came from Russell himself. The paradox he discovered in 1901 — does the set of all sets that do not contain themselves contain itself? — drove Frege's system into contradiction. Russell's letter to Frege in 1902, sent while the second volume was already at the printer, left the German logician with no defence; in an appendix Frege conceded that the foundation of his work had collapsed. Russell's answer, the structural backbone of the Principia, was the theory of types: sets and propositions are arranged into a hierarchy, and no object of a given type may quantify over itself. The restriction blocks the paradox, but whether it is a natural part of logic or an ad hoc patch has been argued ever since.

The book's symbolic density is legendary. The proposition 1+1=2 is proved on page 379 of Volume I, resting on several hundred preceding definitions and theorems. Russell later wrote, "My intellect never quite recovered from the strain." In 1931 Kurt Gödel's incompleteness theorems struck the programme hard: in any sufficiently strong consistent axiom system there will always be true statements that cannot be proved within it. The strictest version of logicism became untenable after that — yet the Principia remained the standard reference frame of mathematical logic for the rest of the twentieth century.

The book's deeper legacy is the tone it set for analytic philosophy. Ludwig Wittgenstein arrived in Cambridge as Russell's student in 1911; in 1921 he dedicated the Tractatus Logico-Philosophicus "to Bertrand Russell and Gottlob Frege." Rudolf Carnap and the Vienna Circle built logical positivism out of this symbolic language; Willard Van Orman Quine's work in the 1940s through 1960s grew from the same soil. Whitehead, after the volumes were finished, left Cambridge for Harvard and veered into the very different territory of process philosophy (Process and Reality, 1929). Russell turned to political activism, popular philosophy, and a half-century public career that ended with the Nobel Prize for Literature in 1950. The Principia is the paradigmatic monument of a programme that could not be finished — and that nonetheless reshaped a discipline irreversibly.