Abstract Singular Terms and Thin Reference

Abstract

The prevailing approach to the problem of the ontological status of mathematical entities such as numbers and sets is to ask in what sense it is legitimate to ascribe a reference to abstract singular terms; those expressions of our language which, taken at face value, denote abstract objects. On the basis of this approach, neo-Fregean Abstractionists such as Hale and Wright have argued that abstract singular terms may be taken to effect genuine reference towards objects, whereas nominalists such as Field have asserted that these apparent ontological commitments should not be taken at face value. In this article I argue for an intermediate position which upholds the legitimacy of ascribing a reference to abstract singular terms in an attenuated sense relative to the more robust ascription of reference applicable to names denoting concrete entities. In so doing I seek to clear up some confusions regarding the ramifications of such a thin notion of reference for ontological claims about mathematical objects.Keywords: reference, abstract singular terms, ontology, abstract objects 1 Charles Parsons (2008, p. 1) captures the standard contemporary meaning of the term "abstract object" when he characterizes it as referring to objects that are "not located in space and time" and which do not "stand in causal relations." Mathematical entities such as numbers, geometrical shapes and sets are thus paradigmatic abstract objects. 2 Although the framework for all three assumptions is Fregean, it is significant that the joint paper by Nelson Goodman and W. V. O. Quine, "Steps towards a Constructive Nominalism" (1947), is the earliest frequently cited paper which frames the problem of "abstract" entities in terms of assumption (ii).ABSTRACT SINGULAR TERMS AND THIN REFERENCE 277