 Mill’s account of mathematics is brief, and raises many issues. Amongst the most pressing questions pertain to the status of the objects which mathematicians talk about. The Platonist can characterize the claims of mathematics as claims about abstract objects—but, as a naturalist, no such option is open to Mill. “All numbers must be numbers of something: there are no such things as numbers in the abstract” (System, VII: 254). Similarly, there are no real objects corresponding to the definitions of geometry (System, VII: 225). Mill, rather, claims that numbers are properties of aggregates and as such denote aggregates with those properties, and takes geometrical objects to be limit cases of real world objects (System, VII: 610–1, 226–9). More, however, needs to be said about the referents of the entities appealed to in the higher regions of mathematics—higher order sets, imaginary numbers, the transfinites—as well as how to construe ‘limit cases’ naturalistically (Shapiro 2000: 91–102, 226–56; Kitcher 1998). One can, perhaps, take mathematical objects to be fictions (Balaguer 2014)—but specifying how such fictions can be subject to constrained standards of truth and falsity remains difficult.