Principia Mathematica

A three-volume work of mathematical logic that tries to show that mathematics can be derived from logic alone, the programme known as logicism. Whitehead and Russell began it after Russell discovered the paradox of the set of all sets that are not members of themselves, which had wrecked Frege's earlier attempt. Their answer is the ramified theory of types, which sorts objects, classes and propositions into levels so that no class can contain itself, supported by the controversial axiom of reducibility; infinity and choice are treated as hypotheses. Using a notation adapted from Peano, the volumes develop propositional and predicate logic, Russell's theory of descriptions, classes and relations, and then cardinal and ordinal arithmetic and the beginnings of real numbers. The proof that one plus one equals two arrives only after hundreds of pages. A planned fourth volume on geometry was never written. Gödel's incompleteness theorems of 1931, framed as results about systems like Principia, showed that no such consistent system can capture all the truths of arithmetic.

It is the founding monument of modern symbolic logic, and knowing what it attempted explains the logic, computing and philosophy of mathematics that followed it.