Looking back from this moment, I think I see things a little differently. The critical question is whether our theoretical description of inquiry gives us a picture that is true to life, preserving the life of inquiry and serving to guide us on its way, or whether it “murders to dissect”, leaving us with nothing but a Humpty Dumpty hodge-podge of false idols and torn and twisted bits of maps that mislead the quest at every turn.
There is a natural semantics that informs mathematical inquiry. It permeates the actual practice even of those who declare for some variety of nominal faith in their idle off-hours. Peirce is unique in his ability to articulate the full dimensionality of mathematical meaning, but echoes of his soundings keep this core sense reverberating, however muted, throughout pragmatism.
If I sift the traditions of theoretical reflection on mathematics according to how well their theoretical images manage to preserve this natural stance on mathematical meaning, I would tend to sort Frege more in a class with Boole, De Morgan, Peirce, and Schröder, since I have the sense when I read them that they are all talking like mathematicians, not like people who are alien to mathematics.