The apparent subject matter of classical mathematics includes such seemingly diverse entities as points, lines, natural numbers, real and imaginary numbers, ordered n-tuples, infinite sequences, functions, vector spaces, topological spaces, groups, rings, and so forth. Working mathematicians have typically proceeded as if these are independently existing ‘Platonic’ entities, and they haven’t worried too much about whether this presupposition is ontologically defensible. But philosophers have often worried about it, and they have explored many positions, most of which may be seen as versions either of realism (Platonism), conceptualism, or nominalism, or as hybrids involving the reduction of entities of some kinds to others. We may think of such philosophical positions as ontological foundations, for at bottom they are claims about what mathematical entities really exist, along with claims about their ultimate natures. For example, noting the reducibility of first-order versions of the various branches of classical mathematics to set theory, it might be held that it is really only sets that comprise the true subject matter of mathematics. Sets would be seen as abstract entities of their own special sort, and the idea that there are also such varieties of entities as numbers, vectors, and functions (etc.) would be abandoned. In this way sets (and set theory) would be seen as the ontological foundation of mathematics. A proponent of this position would owe a principled reason for preferring sets over any other sort of entities (such as properties) to which the apparent mathematical objects might also happen to be reducible. One ingredient of the reason could be the recent ascendency of sets themselves as apparent mathematical entities, but on its own this wouldn’t be very convincing. For if the other apparent entities are up for grabs, why should sets be any different? The important point here is that since any proposed ontological foundation incorporates a claim about the real subject matter of mathematics, it is incompatible with each of its ontological alternatives.
Michael Jubien, “Property-Theoretic Foundations of Mathematics”
