The Contemporary Interest of an old Doctrine

Abstract

My purpose in this talk is to give an overview of the rediscovery of Frege's theorem together with certain of the issues that this rediscovery has raised concerning the evaluation of Frege's logicism—the ‘old doctrine’ of my title.The contextual definition of the cardinality operator, suggested in §63 of Grundlagen— what, after George Boolos, has come to be known as Hume's principle—assertsThe number of Fs = the number of Gs if, and only if, F ≈ G,where F ≈ G (the Fs and the Gs are in one-to-one correspondence) has its usual, second order, explicit definition. The importance of this principle for the derivation of Peano's second postulate (‘Every natural number has a successor’) was emphasized by Crispin Wright (1983, §xix) who presented an extended argument showing that, in the context of the system of second-order logic of Frege's Begriffsschrift, Peano's second postulate is derivable from Hume's principle.