 A considerable obstacle to understanding contemporary philosophical usage of the term, however, is that the notion does not seem to be a unitary one. In the case of abstract objects (the case discussed by Frege) the criterion of identity for Fs is thought of as an equivalence relation holding between objects distinct from Fs. Thus the criterion of identity for directions is parallelism of lines, that is, the direction of line a is identical with the direction of line b if and only if line a is parallel to line b. The criterion of identity for numbers is equinumerosity of concepts, that is, the number of As is identical with the number of Bs if and only if there are exactly as many As as Bs. The relation between the criterion of identity for Fs and the criterion of application for the concept F (the standard for the application of the concept to an individual) is then said by some (Wright and Hale 2001) to be that to be an F is just to be something for which questions of identity and distinctness are to settled by appeal to the criterion of identity for Fs. (Thus, when Frege went on to give an explicit definition of numbers as extensions of concepts he appealed to it only to deduce what has come to be called Hume’s Principle – his statement of his criterion of identity for numbers in terms of equinumerosity of concepts, and emphasised that he regarded the appeal to extensions as inessential.) In the case of concrete objects, however, things seem to stand differently. Often the criterion of identity for a concrete object of type F is said to be a relation R such that for any Fs, x and y, x=y if and only if Rxy. In this case the criterion of identity for Fs is not stated as a relation between entities distinct from Fs and the criterion of identity cannot plausibly be thought of as determining the criterion of application. Another example of the lack of uniformity in the notion of a criterion of identity in contemporary philosophy is, in the case of concrete objects, a distinction customarily made between a criterion of diachronic identity and a criterion of synchronic identity; the former taking the form “x is at t the same F as y is at t′ if and only if…”, where what fills the gap is some statement of a relation holding between objects x and y and times t and t′. (In the case of persons, for example, a candidate criterion of diachronic identity is: x is at t the same person as y is at t′ if and only if x at t is psychologically continuous with y at t′.) A criterion of synchronic identity, by contrast, will typically specify how the parts of an F-thing existing at a time must be related, or how one F at a time is marked off from another.