Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.

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. (shrink)