Perhaps the most popular contemporary alternative to fictionalism is the version of non-naturalist realism offered by the neo-Fregean thesis that abstract mathematical beliefs can be justified as analytic truths that follow from logic and certain meaning stipulations. The idea has been most fully developed in connection with arithmetic, where Crispin Wright has shown how Peano’s postulates can be derived within the framework of second-order logic from nothing except the Humean principle that the same number attaches to equinumerous concepts.  According to Wright, this principle can be viewed as an implicit definition of our concept of number. If this is right, then it has indeed been shown that arithmetic, and therewith the existence of numbers as abstract objects, follows from logic and definition alone (Wright 1983, Hale and Wright 2003). There has also been some attempt to extend the programme to mathematical analysis (Shapiro 2000, Wright 2000).