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 is a three-volume work on the foundations of mathematics by Alfred North Whitehead and Bertrand Russell, published in 1910, 1912, and 1913, with a second edition appearing in 1925–1927 that added a new Introduction, an Appendix A replacing ✱9, and two new appendices. It was meant to follow Russell’s 1903 The Principles of Mathematics, but the authors found this impractical for both philosophical and practical reasons. The work had three goals: to analyze the ideas and methods of mathematical logic as thoroughly as possible while minimizing primitive notions, axioms, and inference rules; to express mathematical propositions in symbolic logic with the most convenient notation; and to resolve the paradoxes that troubled logic and set theory at the turn of the 20th century, such as Russell’s paradox. This last goal led to the adoption of the theory of types, which imposes grammatical restrictions on formulas to prevent the unrestricted comprehension of classes, properties, and functions, making formulas like those that would define the Russell set ill-formed.

Scope of foundations laid

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were absent, but by the end of the third volume, experts could see that much of known mathematics could in principle be developed within its formalism, though they also recognized how lengthy such a development would be. A fourth volume on geometry was planned, but the authors admitted they were intellectually exhausted after finishing the third. The work sparked interest in symbolic logic, popularizing it and demonstrating its power, and the Modern Library ranked it 23rd among the top 100 English-language nonfiction books of the 20th century.

Theoretical basis

Unlike a formalist theory, as Kurt Gödel noted in his criticism, the logicistic theory of the Principia lacks a precise statement of its syntax. The theory embeds the notions of truth and falsity in the concept of a primitive proposition, whereas a pure formalist theory would not assign meaning to its symbols, specifying only how they behave according to grammar.

In the Principia, interpretations are presented in terms of truth-values for symbols like “⊢” (assertion of truth), “∾” (logical not), and “V” (logical inclusive OR). For contrast, a contemporary formal system would start with a set of symbols, build strings by concatenation, define formation rules for well-formed formulas, specify transformation rules (axioms), and include a rule of inference like modus ponens, which detaches a conclusion from premises and discards the premises, proceeding mechanistically by grammar alone. The Principia shares both significant similarities and differences with such a contemporary formal theory.