Principia Mathematica: φ! versus φ

Abstract

Studying the history of mathematical logic in school, on the web or in comics,1 one will surely come upon Whitehead and Russell’s monumental three-volume Principia Mathematica. In the Encyclopedia Britannica we find the following, widely accepted, characterization of the work: Eventually, Russell’s attempts to overcome the paradox resulted in a complete transformation of his scheme of logic, as he added one refinement after another to the basic theory. In the process, important elements of his ‘Pythagorean’ view of logic were abandoned. In particular, Russell came to the conclusion that there were no such things as classes and propositions and that therefore, whatever logic was, it was not the study of them. In their place he substituted a bewilderingly complex theory known as the ramified theory of types, which, though it successfully avoided contradictions such as Russell’s Paradox, was (and remains) extraordinarily difficult to understand. By the time he and his collaborator, Alfred North Whitehead, had finished the three volumes of Principia Mathematica (1910–13), the theory of types and other innovations to the basic logical system had made it unmanageably complicated. Very few people, whether philosophers or mathematicians, have made the gargantuan effort required to master the details of this monumental work. It is nevertheless rightly regarded as one of the great intellectual achievements of the 20th century. (Monk, 2013)