Principia Mathematica

A three-volume work by Alfred North Whitehead and Bertrand Russell that sought to derive mathematics from symbolic logic, using a formal system designed to avoid paradoxes.

Logicism: The philosophical thesis that mathematics is reducible to logic. Whitehead and Russell made this thesis the work’s central program.
Definition: A statement that assigns a precise meaning to a symbol or expression. Whitehead and Russell define mathematical notions through logical vocabulary.
Alfred North Whitehead: An English mathematician and philosopher who co-authored Principia Mathematica with Bertrand Russell. Whitehead collaborated with Russell on the work’s mathematical development and composition.
Gödel's incompleteness theorems: Two theorems showing that sufficiently expressive, consistent formal systems cannot prove every arithmetic truth or establish their own consistency. They constrained hopes that a formal system like Principia Mathematica could settle all foundational questions.
The Principia Mathematica Whitehead–Russell edition: The first edition of Principia Mathematica, published in three volumes from 1910 to 1913. Its structure and notation show the original scope of the authors’ foundational undertaking.
Type theory: A system that classifies expressions into types and restricts how they may be combined. The ramified hierarchy of types blocks self-reference behind logical paradoxes.
Formal proof: A finite sequence of steps in which each conclusion follows from axioms or earlier steps by specified rules. The work presents mathematics as derivations governed by formal inference rules.
Bertrand Russell: A British philosopher and logician who co-authored Principia Mathematica with Alfred North Whitehead. Russell’s logicist program and discovery of a set-theoretic contradiction helped shape the book.
Hilbert's program: David Hilbert’s project to formalize mathematics and prove the consistency of its foundations by finitary means. Its proof-theoretic strategy offered a different response to foundational uncertainty.
Axiom of reducibility: An axiom asserting that every function of a given order is formally equivalent to one of predicative order. It simplified the ramified hierarchy but weakened the claim that mathematics followed from logic alone.

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