Before I can discuss Peirce’s “number of” function in greater detail I will need to deal with an expositional difficulty I have been carefully dancing around all this time, but one which 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 is already to be numbered among the class of dyadic relatives we’ve been dealing with all this time. But Peirce’s manner of representing a dyadic relative term mentions the “rèlate” first and the “correlate” second, a convention going over into functional terms as making the functional value first and the functional argument second. The problem is, almost anyone brought up in our present time frame is accustomed to thinking of a function as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second.
Syntactic wrinkles of this sort can be ironed out smoothly enough in a framework of flexible interpretive conventions, but not without introducing an order of anachronism into Peirce’s text I want to avoid as much as possible. This will require me to experiment with various styles of compromise. Among other things, 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.