Remember the part in Frege’s Grundlagen where he starts to talk about abstraction by talking about the direction of lines? Two lines have the same direction if and only if they are parallel; this gives an identity criterion for directions of lines. Ever wondered why Frege starts bringing in geometry? What the historical context and possible influences were?
Paolo Mancosu has you covered:
I offer in this paper a contextual analysis of Frege’s Grundlagen, section 64. It is surprising that with so much ink spilled on that section, the sources of Frege’s discussion of definitions by abstraction have remained elusive. I hope to have filled this gap by providing textual evidence coming from, among other sources, Grassmann, Schlömilch, and the tradition of textbooks in geometry for secondary schools (including a textbook Frege had used when teaching in a Privatschule in Jena in 1882–1884). In addition, I put Frege’s considerations in the context of a widespread debate in Germany on ‘directions’ as a central notion in the theory of parallels.
Grundlagen, Section 64: Frege’s Discussion of Definitions by Abstraction in Historical Context, History and Philosophy of Logic 36 (1), 2015