Principia Mathematica tried to build mathematics from pure logic
Alfred North Whitehead and Bertrand Russell meant to write a sequel to a single book. Instead the subject swelled into three dense volumes, published between 1910 and 1913, attempting to derive mathematics from a minimal set of logical axioms. A planned fourth volume, on geometry, never came: the authors confessed to intellectual exhaustion.
The project began as a second volume of Russell's 1903 The Principles of Mathematics, but the authors found the field far larger than they had supposed and many earlier puzzles newly solvable. Their introduction set three goals: to reduce mathematical logic to as few primitive ideas, axioms and inference rules as possible; to state mathematical claims precisely in symbolic notation; and to resolve paradoxes, such as Russell's own, that were troubling logic and set theory around 1900.
That last goal produced the theory of types, which imposes grammatical restrictions so that formulas describing problematic objects like the Russell set simply count as ill-formed. The work covered set theory and cardinal, ordinal and real numbers. It stopped short of deeper analysis, yet experts could see that much of known mathematics could in principle be built this way, and also how enormously long such a construction would be.
Its approach differs from modern formal systems. Kurt Gödel criticised its lack of a precise statement of syntax, and Stephen Kleene described it as intuitive axiomatics, axioms meant to be believed as plausible truths rather than symbols shuffled by rules. It brings in truth and falsity almost at once, borrowing Gottlob Frege's definition of a truth-value, and treats inference as something no symbol can fully capture, recorded only by writing down the conclusion.
A second edition in 1925 to 1927 added a new introduction and appendices, reworking the primitive propositions and introducing atomic propositions joined into molecular ones. The books did much to popularise symbolic logic and show its power, and the Modern Library ranked Principia Mathematica 23rd among the century's best English-language nonfiction.
Source: Principia Mathematica
