Frege on the introduction of real and complex numbers by abstraction and cross-sortal identity claims

Abstract

AbstractIn this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers inDie Grundlagen der Arithmetik(1884) and the status of cross-sortal identity claims with side glances atGrundgesetze der Arithmetik(vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context ofGrundlagen. I show why Frege’s strategy in the case of the projected definitions of real and complex numbers inGrundlagenis modelled on his definitional introduction of cardinal numbers in two steps, tentatively via a contextual definition and finally and definitively via an explicit definition. I argue that the strategy leaves a few important questions open, in particular one relating to the status of the envisioned abstraction principles for the real and complex numbers and another concerning the proper handling of cross-sortal identity claims.