Conclusion
In this survey of central issues and proposals in the philosophy of mathematics, we saw with regard to the issue of the ontology of mathematics how Platonists and nominalists have offered importantly different conceptions.
Platonists had the significant advantage of being able to take mathematical discourse literally; that is, they did not have to rewrite mathematical (and scientific) theories to avoid commitment to mathematical objects. With the possible exception of deflationary nominalism, all of the other nominalist views had to create a parallel discourse to accommodate mathematics. That is, they had to provide a reformulation of the theories in question in a nominalistically acceptable language (either by using a fiction operator or a suitable modal language).Platonists, in turn, faced a significant challenge to make sense of the epistemological issue of how we can have knowledge of abstract mathematical objects and structures to which we have no causal access. As we saw, various epistemological strategies were devised. From reconstructive accounts in which mathematical objects emerge from logical concepts (in Fregean Platonism) through accounts based on the intuition of basic mathematical facts (in Godelian Platonism) to the use of concrete templates as a vehicle to our knowledge of abstract patterns (in structuralist Platonism), Platonists have spent significant resources trying to make sense of mathematical knowledge. This was not a problem, however, that the nominalist faced. Bluntly put, if mathematical objects do not exist, we do not have to provide an account of how we come to know them.
However, understanding the application of mathematics then becomes a significant issue for the nominalist. If mathematical objects do not exist, how can we understand the success of the application of mathematics to the physical world? Nominalists addressed this issue directly, devising strategies to explain this success despite the non-existence of mathematical objects. They emphasized that mathematical theories need not be true to be good, as long as they are conservative (in mathematical fictionalism), or highlight the role played by possible structures in the application of mathematics (in modal structuralism). But for the deflationary nominalist, the problem of the application of mathematics is just an artifact - a philosophical creation of something that is not a philosophical issue (Azzouni 2000). Applied mathematics, just as its pure counterpart, involves finding out what follows from what. The trouble is that it is often not transparent what should be the consequences in each case, particularly those involving premises that describe aspects of the physical world. In the end, it does not look like we have here a special philosophical issue after all.