Is Mathematics the Theory of Instantiated Structural Universals?

The paper contends that one cannot defend realism about numbers on the basis of a metaphysical realism about instantiated structural universals, suggesting that it is misleading to take a. metaphysical view as a basis for the ontology and epistemology of mathematics. The author criticizes Bigelow's attempt to reduce number theory to a metaphysical theory about instantiated structural universals, which purports to reduce number theory to the theory of structural universals, and which flies in the face of solid mathematical knowledge. The study begins with a presentation of Armstrong's theory of structural properties as instantiated universals and of Lewis's devastating criticism of this theory, arguing that several responses to this criticism, by Armstrong, Bigelow, and more recently, by Joan Pages, can hardly succeed. Finally, it contends that one possible construal of structural universals via non-well-founded sets is resisted by the realist structuralist about mathematics. The conclusion highlights an issue that would have to be addressed by anyone who wants to pursue Bigelow's reductionist project: the alleged countability of the real numbers.