Principia Mathematica was published in three volumes by Cambridge University Press: Volume I in 1910, Volume II in 1912, and Volume III in 1913. Written by Bertrand Russell and Alfred North Whitehead over a period of nearly ten years, it remains one of the most ambitious intellectual undertakings of the twentieth century: an attempt to show that all of mathematics can be derived from a small set of logical axioms and inference rules, without appeal to intuition, self-evidence, or any extra-logical concepts.
The project emerged from Russell’s work on the foundations of mathematics in the early 1900s, particularly his discovery of Russell’s Paradox (1901), which showed that naive set theory was inconsistent. If the set of all sets that do not contain themselves both does and does not contain itself, then mathematics built on set theory rests on contradiction. Principia Mathematica was Russell and Whitehead’s response: a reconstruction of mathematics from the ground up, using a type theory that avoided the paradox by organizing sets into a hierarchy of “types.”
The work is written almost entirely in symbolic notation — the famous proof that 1+1=2 does not appear until page 379 of Volume I, and even that proof requires several hundred pages of prior definitions and theorems. The notation is Russell and Whitehead’s own invention and is now largely superseded by more modern systems, which makes the original text difficult to read even for trained logicians.
The intellectual legacy of Principia Mathematica is paradoxical. Its specific program — logicism, the reduction of mathematics to logic — is generally considered to have failed, definitively undermined by Gödel’s incompleteness theorems (1931), which showed that any consistent formal system powerful enough to express basic arithmetic must contain truths that cannot be proved within the system. But the methods and concepts developed in Principia — formal logic as a tool for philosophical analysis, the theory of types, the notation and techniques of mathematical logic — became the foundation of analytic philosophy and of theoretical computer science.
Collecting Principia Mathematica
First edition (Cambridge University Press, 1910–1913): Three volumes, blue cloth.
Market values:
- First edition, complete set, fine: $30,000–$80,000
- Very good set: $15,000–$40,000
- Volume I alone, very good: $5,000–$15,000
- Second edition (1925–1927): $2,000–$5,000
This is one of the most valuable scientific works of the twentieth century. Complete first-edition sets in good condition are genuinely rare; the print run was small (fewer than 750 copies of Volume I), and the books were heavily used by the mathematicians and logicians who owned them. Cambridge University Press reportedly lost money on the publication, and Russell and Whitehead each contributed £50 toward the printing costs.