Abstract objects and semantics: An essay on prospects and problems with abstraction principles as a means of justifying reference to abstract objects

Abstract

The aim of this thesis is to reconsider the role that abstraction principles play, for neo-Fregeans, in establishing reference to abstract objects, in a way that brings to light both their significance and their limitations that have not been fully recognized in the literature so far. I argue, on the one hand, that, on the neo-Fregean account of securing reference to abstract object,abstraction principles play a crucial role when it comes to establishing abstract reference.In other words, I believe that the reference of an abstract term requires a formulation of an appropriate abstraction principle. I argue that both defenders and critics of neo-Fregeanism who fail to recognize the role that abstraction principles play in reference-fixing misunderstand the view. In particular, I argue that abstraction principles enable us to fix identity conditions for the abstract objects in question and that by doing so enable us to distinguish cases where we succeed in fixing reference via abstraction and cases where we don’t. I argue that,while numerical terms or such terms as ‘direction’ meet this restriction, there are many, for example, fictional terms,that do not –because of lack of appropriate criteria of identity. However, on the other hand, I also believe that there is an important obstacle to fixing reference via abstraction principles that even abstraction principles usually taken to be in good standing, those used to introduce numerical terms or ‘directions’, encounter. Roughly speaking, this obstacle is raised by the status of the equivalences involved in abstraction principles and a question of whether a nominalist could ever accept that such an equivalence holds. I argue that a careful consideration of the nominalist’s perspective reveals one general, but fundamental consequence –namely, that the appeal to abstraction principles assumes the very platonism they are meant to justify. Altogether, the above claims form the core of the following reasoning. By providing identity conditions for the abstract objects in question, abstraction principles enable us to distinguish between cases where fixing reference of abstract terms is successful and cases where it is not –while many terms, such as fictional terms, are excluded by it, some, for instance, ‘numbers’ or ‘directions’, are preserved. However, if we take into account the above general worry, raised by taking into consideration the perspective of the nominalist,then even abstraction principles usually taken to be in good standing, such as those that introduce numerical terms and ‘directions’ are threatened. Unless, or until, this worry is explained away,we are left with skepticism about abstract reference