Principia Mathematica

Lore & Background

Moreover, on many fundamental questions which had been left obscure and doubtful in the former work, they arrived at what they believed to be satisfactory solutions. PM had three aims: to analyse to the greatest possible extent the ideas and methods of mathematical logic and to minimise the number of primitive notions, axioms, and inference rules; to precisely express mathematical propositions in symbolic logic using the most convenient notation; and to solve the paradoxes that plagued logic and set theory at the turn of the 20th century, like Russell's paradox. This third aim motivated the adoption of the theory of types, which adopts grammatical restrictions on formulas that rule out the unrestricted comprehension of classes, properties, and functions.

Reader's Guide

The Principia Mathematica covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism. A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third. PM sparked interest in symbolic logic and advanced the subject, popularizing it and demonstrating its power. The Modern Library placed PM 23rd in their list of the top 100 English-language nonfiction books of the 20th century. As noted in the criticism by Kurt Gödel, unlike a formalist theory, the 'logicistic' theory of PM has no precise statement of the syntax of the formalism. PM embeds the notions of truth and falsity in the notion of primitive proposition, unlike a pure formalist theory that would not provide the meaning of the symbols.

Did You Know?

• The theory of types in PM was adopted to solve paradoxes like Russell's paradox.
• A fourth volume on geometry was planned but abandoned due to the authors' intellectual exhaustion.