Structure and Abstraction
Ave Caesar! Morituri te salutant.
My ontological inclinations are nominalistic, and nominalism traditionally faces a number of stern challenges. One is from the theory of predication and universals. I think this challenge is, for a philosophical one, fairly easy for a nominalist to face. A far more daunting prospect is to reconcile the nominalist's denial of the existence of abstract objects with the acceptance of a vast body of mathematical propositions by a large number of highly intelligent people. Taken at face value, the mathematical propositions these people hold true entail the existence of untold infinities of abstract mathematical objects.
However, rather than presuppose or work within a nominalist framework I shall be exploring aspects of the kind of Platonistic realism of mathematical objects which I consider the most promising and plausible, namely structuralism. I shall not attempt to argue for structuralism against other kinds of Platonistic mathematical ontology. Rather I wish to canvass the possibility of a union--some might consider it a shotgun wedding-- between the philosophies of mathematics of Dedekind and Frege. This is not in order to rescue the logicism they held in common, but because each emphasizes in his philosophy of mathematics something the other can profitably use, namely structure ( Dedekind) and abstraction ( Frege).
It may seem anachronous to take abstraction as the Fregean component of the proposed synthesis because after all Frege vociferously rejected and indeed poured scorn on what other mathematicians such as Dedekind and Cantor and various empiricists called 'abstraction'. By 'abstraction' I shall intend the procedure of contextual introduction of singular terms and their