Comment by griffzhowl

That sounds like the same circularity, since you'll have to assume numbers exist before proving any statements about them.

Physical objects aren't like that because you can discover that they exist by empirical investigation.

In mathematics the discoveries are about the logical implications of sets of axioms. Some of those axioms contain assertions of existence, like a number 0 in Peano arithmetic or the empty set in set theory, and then you can prove statements about these objects based on the axioms. It's circular to infer from these conclusions that the axioms are true.

What's interesting is why certain axiom systems are so useful and fruitful. Personally I think it's because they evolved that way from our investigations of the physical world, but that's another matter