Principia Mathematica
Alfred North Whitehead and Bertrand Russell's Principia Mathematica ambitiously seeks to ground all mathematics in formal logic via the logicist thesis, constructing arithmetic from primitive propositions and logical types to resolve paradoxes like Russell’s own vicious circle principle, which bans self-referential sets (e.g., "the set of all sets"). The ramified theory of types stratifies propositions and functions into hierarchies to avoid circularity, while propositional functions (e.g., "x is a number") replace classes with intensional definitions. The system employs PM notation—dots for logical connectives—to rigorously derive theorems like 1+1=2 after hundreds of pages. Though incomplete (Gödel later proved its incompletability) and criticized for axiom of reducibility ad hocness, it revolutionized formal logic, inspired computability theory (Turing’s work), and remains a landmark in analytic philosophy and foundations of mathematics, despite its practical eclipse by ZFC set theory.
