I'm trying to understand Frege's argument for the existence of mathematical objects. Specifically, I'm trying to understand the premises of classical semantics and truth.
Classical Semantics. The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects.
Truth. Most sentences accepted as mathematical theorems are true (regardless of their syntactic and semantic structure).
First, doesn't classical semantics presuppose the existence of mathematical objects? Frege's argument ends at the conclusion that there exist mathematical objects, but this seems to be circular reasoning since classical semantics states that singular terms of mathematics refer to mathematical objects. (Perhaps I should be thinking of it less like an existential and more like a hypothetical. Does classical semantics merely state that mathematical statements purport to refer to mathematical objects without asserting the existence of mathematical objects?)
Now, for truth: why does it only assert that most theorems are true? In order for a sentence to be accepted, wouldn't a proof have to exist? Does Frege just mean that there are some theorems that we believe to be true but aren't true?