The kinds of mathematical objects

The Kinds of Mathematical Objects is an exploration of the taxonomy of the mathematical realm and the metaphysics of mathematical objects. I defend antireductionism about cardinals and ordinals: the view that no cardinal number and no ordinal number is a set. Instead, I suggest, cardinals and ordinals are sui generis abstract objects, essentially linked to specific abstraction functors (higher-order functions corresponding to operators in abstraction principles). Sets, in contrast, are not essentially values of abstraction functors: the best explanation of the nature of sethood is given by a variation on the standard iterative account. I further defend the theses that no cardinal number is an ordinal number and that the natural numbers are, as Frege maintained, all and only the finite cardinal numbers. My case for these conclusions relies not on the well-known antireductionist argument developed by Paul Benacerraf, but on considerations about ontological dependence. I argue that, given generally accepted principles about the dependence of a set on its elements, ordinal and cardinal numbers have dependence profiles that are not compatible with any version of set-theoretic ontological reductionism. In addition, a formal framework for set theory with sui generis abstract objects is developed on a type-theoretical basis. I give a philosophical defence of the choice of type theory and discuss various questions relating to the nature of its models.