One form of structuralism says that mathematics is just the proving of theorems from axioms, which may or may not hold of (or approximate) various real systems. Something like this is clearly right for things like topology, group theory, and the like – when we prove something about topological spaces (or better yet, about compact hausdorff spaces), we don’t mean to prove it of any particular thing, but rather just know that whenever we find some entity that satisfies the axioms of topology (plus the compact hausdorff axioms), the theorem will be true of this entity. Whether or not there is such an entity is almost a side concern, though the fact that they seem to arise in various areas of mathematics convinces us that the theorems are likely to be useful.
The structuralist says that this is the case with Peano arithmetic, and set theory as well. We’re not proving anything about actual numbers or sets (like the platonist claims) – we’re just proving theorems that will hold of any entities that happen to satisfy the axioms we use. Something like this seems to be the position of Saunders Mac Lane in his 1997 paper “Despite Physicists, Proof is Essential in Mathematics” (Synthese 111:147-154),
However, this position seems problematic, because it doesn’t explain how we are justified in many of our applications of mathematics. (Platonism and fictionalism don’t obviously work much better, but I think I can sketch an account of how they at least partially justify us in these applications.) If our theorems about real numbers are just theorems about whatever happens to satisfy the real number axioms, rather than about anything independently existing entities, then to apply the theorems, we would need to show that the system we’re applying it to satisfies the axioms. So in order to apply the intermediate value theorem to distances, we’d need to show that distances are dense, linearly ordered, have no endpoints, and are topologically complete. In order to apply them to probabilities, temperatures, electric charges, masses, times, and the like, we’d need to show that these axioms apply to each of those domains. However, it seems unlikely to me that we’ve actually shown that any of these quantities satisfies the real number axioms, much less all of them.
The platonist has a way out by saying that (somehow) we’ve discovered facts about these entities we call real numbers, which aren’t the physical entities we’re talking about. Instead, we see that in each of these physical situations, our investigation reveals that there is some sort of abstract entity involved, and we can use inductive reasoning to suggest that the real numbers are the appropriate ones. Thus, even if we can’t directly support the infinite divisibility, or the topological completeness, of any of these realms, we might be able to inductively support them by inferring that the reals (a structure we already know about) would give the best explanation of the phenomena we’re observing. If we had to restrict ourselves to a set of structural axioms that we knew to hold, we’d have to convince ourselves that some closely related set of axioms wouldn’t be better. If the platonist has a reason for thinking that the real numbers are a special set of entities, then we could infer this convergence on one set of axioms without having to establish it completely independently each time.