How does metamathematical formalism interpret statements such as “4 is even”?

In the context of metamathematical formalism, mathematical statements are not interpreted as referring to abstract entities or real-world objects. Instead, this view holds that ordinary mathematical sentences are actually about the formal sentences and theories themselves.1

Under this interpretation, the statement “4 is even” is not taken to mean that the number 4 possesses the property of being even in a literal or ontological sense. Rather, it signifies that the specific sentence “4 is even” is a theorem that follows from the axioms of arithmetic.1

Metamathematical formalism, developed by Haskell Curry, emphasizes the syntactic manipulation of symbols within a formal system rather than focusing on the meanings or referents of those symbols. By treating mathematical discourse as a system of rules for manipulating formulas, formalists aim to secure the reliability of mathematics through the study of its syntax, independent of any external interpretation.123