Full Idea
In his middle period Frege rated identity indefinable, on the ground that every definition must take the form of an identity-statement. Frege introduced the notion of criterion of identity, which has been widely used by analytical philosophers.
Clarification
His middle period is 1891-1906
Gist of Idea
Frege introduced criteria for identity, but thought defining identity was circular
Source
Michael Dummett (Frege philosophy of mathematics [1991], Ch.10)
Book Reference
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.113
A Reaction
The objection that attempts to define identity would be circular sounds quite plausible. It sounds right to seek a criterion for type-identity (in shared properties or predicates), but token-identity looks too fundamental to give clear criteria.