Principia Mathematica is a three-volume work on the logical foundations of mathematics, authored by Alfred North Whitehead and Bertrand Russell and published between 1910 and 1913. The work attempts to derive all of classical mathematics from a small set of logical axioms and inference rules, using a formal symbolic language developed in part from Gottlob Frege's predicate logic and Giuseppe Peano's notation. It is among the most ambitious formal projects in the history of logic and philosophy of mathematics, and its influence extends into mathematical logic, analytic philosophy, computer science, and the philosophy of language.
The central project of Principia Mathematica is logicism - the thesis that mathematics is reducible to logic. To accomplish this, Whitehead and Russell constructed an elaborate type-theoretic hierarchy intended to block the paradoxes, particularly Russell's Paradox, that had destabilized naive set theory. The system employs the theory of types to prevent self-referential constructions, and introduces ramified type theory along with the controversial Axiom of Reducibility, which many logicians have regarded as an ad hoc concession that undermines the purity of the logicist program. The work also introduces the no-classes theory, treating classes as logical fictions rather than genuine objects.
In 1931, Kurt Gödel published his incompleteness theorems, demonstrating that any consistent formal system expressive enough to capture basic arithmetic cannot prove all truths expressible within it, and cannot prove its own consistency. This result is generally taken to place fundamental limits on the Principia program, though the precise implications for logicism as a philosophical thesis remain a matter of debate. Separately, Fregean logicism had already been severely undermined by Russell's discovery of the paradox in Frege's Grundgesetze; Principia represents the most sustained attempt to rebuild it. Later developments in set theory, particularly the Zermelo-Fraenkel axiom system, offered an alternative foundation that most working mathematicians adopted in preference to the type-theoretic framework of Principia.
The notation and methods of Principia Mathematica were foundational to the development of formal logic in the twentieth century, influencing Wittgenstein's early work, the Vienna Circle's program of logical empiricism, and the emergence of theoretical computer science through figures such as Alan Turing and Alonzo Church.
There is broad consensus in mathematical logic that Gödel's incompleteness theorems constrain any program of the type Principia Mathematica represents; see Mathematical Logic Consensus. Whether this refutes logicism as a philosophical position, or merely requires its revision, is not settled by that consensus and is addressed on the Debate page.