Logic Seminar: Session 9

Neo-Fregeanism in the philosophy of arithmetic is the view that numbers can be defined using a certain so-called “abstraction principle.” This principle (which has been dubbed “Hume’s Principle”) asserts that the number of Fs is identical to the number of Gs iff the Gs can be put into 1-1 correspondence with the Fs. There has been a very vigorous discussion for the last few decades about whether this principle is sufficient to define the numbers, and the more general question of whether definition by abstraction is an acceptable form of definition (and, if so, when.) It is virtually unknown, however, that Bertrand Russell, in the very first months after he developed his logic of relations and attempted to use it to prove logicism (in October 1900), defined the cardinal numbers (and many other mathematical entities) using principles of abstraction. Though Russell developed this “Neo-Fregean” version of logicism, and its philosophical defense, in great detail, virtually none of this work was published, since Russell had come to reject principles of abstraction on philosophical grounds by March 1901. However, the early drafts of his paper “The Logic of Relations” and most of his book The Principles of Mathematics were written while Russell was committed to abstraction principles. (In fact, chapters XIX, XXVI, and XXIX of Principles, first composed in November 1900 and left virtually unchanged in the final published version, include a philosophical defense and widespread application of abstraction principles.) In this talk, Heis reconstructs Russell’s Neo-Fregean phase, the factors that influenced him to defend definition by abstraction, and the arguments that finally led him to abandon it. What emerges is a philosophically extremely rich development, with Russell in some ways anticipating the defenses and objections to Neo-Fregeanism that were independently discovered by philosophers a century later.

 Chair for Session: TBA