From this perspective, the demand for a compositional fictionalist semantics for mathematical sentences is misplaced for the reason that the fictionalist does not propose a different semantics for mathematical sentences at all.
The fictionalist does not make room for modal ignorance to the same extent the realist does.
Suppose that a modal fictionalist does accept an ontology of propositions rich enough to provide for maximal consistent collections of such propositions.
After all, the point of providing a nominalistic proof of the conservativeness of set theory is to show that, without recourse to platonist mathematics, the mathematical fictionalist is able to establish that mathematics is conservative.
In order to establish the conservativeness of mathematics, the mathematical fictionalist has used metalogical results, such as the completeness and the compactness of first-order logic (Field 1992, 1980 , 1989.
Unlike paraphrase nominalists, fictionalists admit that the platonist's face-value semantics for mathematical discourse is correct; but because fictionalists don't believe in abstract objects, they think that mathematical sentences like '3 is prime' are not true.
Given that mathematical objects do not exist, on the mathematical fictionalist perspective, the problem of how we can obtain knowledge of them simply vanishes.
How should the mathematical fictionalist approach physical theories, such as perhaps string theory, that do not seem to be about concrete observable objects?
The problem, however, is that it is not at all obvious that, at least in the form articulated by Field, the mathematical fictionalist program can be extended to set theory.
Similarly to the epistemological problem, the problem of the application of mathematics is partially solved by the mathematical fictionalist.
