  Both of the steps of Benacerraf’s argument are controversial. The first step—that natural numbers have only structural properties—has recently been defended by a variety of mathematical structuralists (Parsons 1990, Resnik 1997, and Shapiro 1997). But this step is denied by logicists and neo-logicists, who claim that the natural numbers are intrinsically tied to the cardinalities of the collections that they number. And the second step—that there can be no objects with only structural properties—is explicitly rejected by all of the structuralists who defend the first step. (For some voices sympathetic to the second step, see Hellman 2001 and MacBride 2005. See also Linnebo 2008 for discussion.)