There is a further subtlety here. As I discussed in “Syntacticism”, in Gödel’s theorems number theory is in fact talking about “number theory”, and we apply a metatheory to prove that “number theory is “number theory”″, and think we’ve proved that number theory is “number theory”. The answer I came to was to conclude that number theory isn’t talking about anything (ie. ascription of semantics to mathematics does not reflect any underlying reality), it’s just a set of symbols and rules for manipulating same, and that those symbols and rules together embody a Platonic object. Others may reach different conclusions.
There is a further subtlety here. As I discussed in “Syntacticism”, in Gödel’s theorems number theory is in fact talking about “number theory”, and we apply a metatheory to prove that “number theory is “number theory”″, and think we’ve proved that number theory is “number theory”. The answer I came to was to conclude that number theory isn’t talking about anything (ie. ascription of semantics to mathematics does not reflect any underlying reality), it’s just a set of symbols and rules for manipulating same, and that those symbols and rules together embody a Platonic object. Others may reach different conclusions.