Numbers Can Be Just What They Have To

structures will not be sets. Rather, speaking of the natural numbers as an abstract structure will be a faqon de parler for the properties common to all progressions. There can be similar treatment of other structures, say the real numbers or whatever. This idea is widely influential under the name of "struc- turalism" but remains problematic in its particulars. (Compare Hellman 1989, Parsons 1990, Resnik 1981 and 1982 and Shapiro 1983.) Another point of view, though, says abstract structure is subtle but not so complex, and the irrelevant features of ZF sets are just technicalities. In fact, the structuralist program is already fulfilled or obviated, depending on how you look at it, by categorical set theory (first described in Lawvere 1964). Sets and functions in this theory have only structural properties. There is no need and no