Full Idea
Having rightly perceived that the fundamental class here was statements of identity between directions, Frege leapt to the conclusion that the basis for introducing new abstract terms consisted of determining the truth-conditions of identity-statements.
Gist of Idea
From basing 'parallel' on identity of direction, Frege got all abstractions from identity statements
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §64-68) by Michael Dummett - Frege philosophy of mathematics Ch.18
Book Reference
Dummett,Michael: 'Frege: philosophy of mathematics' [Duckworth 1991], p.232
A Reaction
This seems to be the modern view - that abstraction consists of the assertion of an equivalence principle. Dummett criticises Frege here (see Idea 9882). There always seems to be a chicken/egg problem. Why would the identity be asserted?
Related Idea
Idea 9882 You can't simultaneously fix the truth-conditions of a sentence and the domain of its variables [Dummett on Frege]