Despite the substantial differences between nominalism and platonism, they have at least one feature in common: both come in many forms. There are various versions of platonism in the philosophy of mathematics: standard (or object-based) platonism (Gödel 1944, 1947; Quine 1960), structuralism (Resnik 1997; Shapiro 1997), and full-blooded platonism (Balaguer 1998), among other views. Similarly, there are also several versions of nominalism: fictionalism (Field 1980, 1989), modal structuralism (Hellman 1989, 1996), constructibilism (Chihara 1990), the weaseling-away view (Melia 1995, 2000), figuralism (Yablo 2001), deflationary nominalism (Azzouni 2004), agnostic nominalism (Bueno 2008, 2009), and pretense views (Leng 2010), among others. Similarly to their platonist counterparts, the various nominalist proposals have different motivations, and face their own difficulties. These will be explored in turn. (A critical survey of various nominalization strategies in mathematics can be found in Burgess and Rosen (1997). The authors address in detail both the technical and philosophical issues raised by nominalism in the philosophy of mathematics.)