 Suppose, first of all, that one wished to deny the existence claim which is a component of platonic realism about arithmetic. One way to do this would be to propose an analytic reduction of talk seemingly involving abstract entities to talk concerning only concrete entities. This can be illustrated by considering a language the truth of whose sentences seemingly entails the existence of a type of abstract object, directions. Suppose there is a first order language L, containing a range of proper names ‘a’, ‘b’, ‘c’, and so on, where these denote straight lines conceived as concrete inscriptions. There are also predicates and relations defined on straight lines, including ‘ … is parallel to …’. ‘D( )’ is a singular term forming operator on lines, so that inserting the name of a concrete line, as in ‘D(a)’, produces a singular term standing for an abstract object, the direction of a. A number of contextual definitions are now introduced: