Fictionalism's Ontological Commitment
Fictionalism about a certain discourse is the view that statements belonging to this discourse are to be interpreted like statements in fictional discourse.
Now as Brian has observed, on the common account of fictional discourse, "Fictional(Fa)" implies "(Ex)Fictional(Fx)" (even though it normally doesn't imply "(Ex)Fx"). So one might think that on the common account, fictionalism can't do with fewer entities than realism, even though it can do with different entities. However, the common account is not committed to "Fictional(a != b)" implying "(Ex)(Ey)(x != y)". After all, it usually allows for "(Ex)(Fictional(Fx) and Fictional (not-Fx))", so why not allow for "(Ex)(Fictional(x != b) and Fictional(x = b))"? So maybe one could endorse fictionalism about mathematics and the common account of fictional discourse without being committed to an infinity of entities by claiming that all the "numbers" talked about in mathematics are in fact identical.
On my own preferred account of fictional discourse, fictionalism about mathematics is in real danger of collapsing into realism. On this account, "Fictional(p)" implies "Possible p", so if, as in mathematics, we have "Fictional(Necessary p)", we get "Possible(Necessary p)", which, if the modalities are the same and obey S5, implies "Necessary p". So perhaps I am both a fictionalist and a platonist about mathematics.
[Update 2003-06-27: The wording of that last paragraph is wrong (as Karl Schafer noticed). Of course fictionalism about mathematics is not "in real danger of collapsing into realism" because fictionalists can still deny either that the necessity of mathematical facts belongs to the fiction, or that the necessity is of the same kind as the possibility of fictions, namely "conceptual", as opposed to merely metaphysical. (Of course the common strategy for fictionalists is instead to reject my preferred account of fictionality.) What I wanted to say is that one could coherently be both a fictionalist and a platonist about mathematics, and hence that "fictionalism" is a somewhat strange name for anti-platonism.]