Higher‐Order Fictionalism and the Infinite Regress

Fictionalism is claimed to allow its advocates to do mathematics without having to commit thereby to the existence of abstract entities. Nevertheless, fictionalism is subject to a charge that has not received enough attention yet, namely, the nominalistic impurity objection. According to this objection, the fictionalist's tenets surreptitiously involve purported reference to abstracta and quantification over abstracta, and, consequently, the fictionalist is inadvertently committed to precisely those entities, which she wanted to reject. In this paper, I argue that the nominalistic impurity objection remains undefeated, and constitutes, for the fictionalist, a significant threat that demands serious attention.