The publication of the three volumes of Principia Mathematica (Cambridge, 1910, 1912 and 1913) was a landmark in the history of mathematical logic. The explicit goal was to derive all mathematical truths from a specific set of axioms and rules of inference designed to avoid the well-known paradoxes that plagued earlier attempts.
There were however many problems from the beginning. To begin with, some axioms, such as the axiom of infinity and the axiom of choice, didn’t look at all like axioms of logic but had to be added. Secondly, the questions regarding the consistency of the theory and its completeness were left unanswered in the Principia. Both questions were addressed by Kurt Gödel, who in 1930 answered the first question for propositional logic in the affirmative. Then, in his landmark paper of 1931, he answered the second question negatively: no formal system large enough to contain elementary arithmetic can be complete.
As it is usually the case, mathematicians sometimes forget some of this history. If we go to our local library and look at any of the several editions, complete or abridged, of Principia Mathematica, we are soon turned off by its merciless notation, little of which resembles any of our standard symbols. Today we have more pedagogically designed textbooks on mathematical logic, some of them covering more ground than that of Principia. Nevertheless, for the section on propositional calculus they follow closely the paths first taken in Principia, save for notation. We also know, and cherish, Gödel’s groundbreaking theorems. But let’s not forget that they were directly inspired by a close reading of Principia.
This brings us to the book under review. Indeed, there are several versions of Principia Mathematica, and it is the second edition of 1925 and 1927 that is being looked at closely by the author. The main changes in the second edition were additions: a new introduction, three appendices, and a list of definitions, a grand total of 66 pages. Minor changes, besides the correction of some typographical errors, include the adoption of the “Sheffer stroke” (“not-both”) to replace the familiar logical connectives “or” and “not” (nowadays a standard exercise in logic textbooks).
But there are two fundamental changes in the second edition of Principia. The first one is an attempt by Russell to address the issues regarding the adoption of the axiom of reducibility in the first edition, widely criticized by many of Russell’s contemporaries. The remedy adopted by Russell in Appendix B of the second edition of Principia was flawed, however, and it was again Gödel who showed that the derivation of the principle of mathematical induction without using the axiom of reducibility is impossible in certain systems. The second fundamental change is the adoption of extensionality: all functions of propositions are truth-functions, and Appendix C is devoted to defend this doctrine.
Linsky’s monograph highlights the main points in all these changes, putting them in context and with regards to the evolution of mathematical logic in the twentieth century. Moreover, this monograph makes available for the first time transcripts of the original manuscripts that Russell wrote for the second edition of Principia, both those that were included in the published version and some that were left out.
The book succeeds in calling our attention to the fundamental changes made to the second edition of Principia, both in its published form and in its unfinished manuscripts. It allows the reader a more balanced reading of this sometimes-neglected contribution to symbolic logic, both in its accomplishments and in its failures.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.