Neo-Fregeanism: An Embarrassment of Riches

Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume’s Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Crispin Wright settling on a another one– stability– as the best bet for neo-Fregeans. However an analogue of the embarrassment of riches objection resurfaces in the meta-theory and I conclude by arguing that the neo-Fregean program, at least insofar is it includes a platonistic ontology, is fatally wounded by it.