Callum wrote:

I think the chess analogy crucially rests on the argument that there are many formal axiomatic systems. In [w]hat sense they already exist?

They exist in potentiality. Like possible worlds, probability and conceivability.

Calhoun wrote:

"In his existential detective story, Why Does the World Exist?, the American philosopher Jim Holt, explains the nominalist view of mathematics:

All of mathematics can be seen to consist of if-then propositions: if such-and-such structure satisfies certain conditions, then that structure must satisfy certain further conditions. These if-then truths are indeed logically necessary. But they do not entail the existence of any objects, whether abstract or material. The proposition "2 + 2 = 4," for example, tells you that if you had two unicorns and you added two more unicorns,then you would end up with four unicorns. But this if-then proposition is true even in a world that is devoid of unicorns—or, indeed, in a world that contains nothing at all. (181)

I haven't read that book but looking at this passage I think there must be more to it than you quote, because this doesn't even remotely sound like an argument for nominalism, absent further argumentation all this seems to suggest is that numbers can be reduced to propositions, but how does that lead to nominalism? Realism isn't just realism about numbers. it also includes possible worlds, universals, propositions etc..

I guess you could be a realist with regards to some universals and propositions but not others. Would the be examples of neccessary propositions that aren't mathematical?