Before I can discuss Peirce’s “number of” function in greater detail I will need to deal with an expositional difficulty that I have been very carefully dancing around all this time, but one that will no longer abide its assigned place under the rug.
Functions have long been understood, from well before Peirce’s time to ours, as special cases of dyadic relations, so the “number of” function itself is already to be numbered among the types of dyadic relatives that we’ve been explicitly mentioning and implicitly using all this time. But Peirce’s way of talking about a dyadic relative term is to list the “relate” first and the “correlate” second, a convention that goes over into functional terms as making the functional value first and the functional argument second, whereas almost anyone brought up in our present time frame has difficulty thinking of a function any other way than as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second.
All of these syntactic wrinkles can be ironed out in a very smooth way, given a sufficiently general context of flexible enough interpretive conventions, but not without introducing an order of anachronism into Peirce’s presentation that I am presently trying to avoid as much as possible. Thus, I will need to experiment with various styles of compromise formation.
The interpretation of Peirce’s 1870 “Logic of Relatives” can be facilitated by introducing a few items of background material on relations in general, as regarded from a combinatorial point of view.