Quine Willard. Ontological reduction and the world of numbers

Article published in «The Journal of Philosophy» — 1964 — Volume LXI — No. 7 (March) — pp. 209-216One conspicuous concern of analytical or scientific philosophy has been to reduce some notions to others, preferably to less putative ones. A familiar case of such reduction is Frege's definition of number. Each natural number n became, if I may speak in circles, the class of all w-member classes. As is also well known, Frege's was not the only good way. Another was von Neumann's. Under it, if I may again speak in circles, each natural number n became the class of all numbers less than n. In my judgment we have satisfactorily reduced one predicate to others, certainly, if in terms of these others we have fashioned an open sentence that is coextensive with the predicate in question as originally interpreted; i.e., that is satisfied by the same values of the variables. But this standard does not suit the Frege and von Neumann reductions of number; for these reductions are both good, yet not coextensive with each other.