How did formalism interpret ordinary mathematical statements?

In the philosophy of mathematics, formalism interprets ordinary mathematical statements by shifting the focus from their potential meanings or references to real-world entities toward the rules governing their manipulation as symbols.12

Formalists contend that mathematical symbols are the primary objects of thought, and they treat mathematics as a system of formulas that can be manipulated without regard to any external meaning. A specific version of this approach, known as metamathematical formalism, posits that ordinary mathematical sentences—which appear to describe objects like numbers—are actually statements about mathematical theories and their syntax.13

For example, under this interpretation, the statement "4 is even" is not taken to literally describe a property of the number 4. Instead, it is understood to mean that the sentence "4 is even" is a provable consequence of the established arithmetic axioms. By concentrating on the formal language and syntax of mathematics, formalists aimed to secure a rigorous foundation for the field, viewing mathematical truth as a matter of provability within a consistent formal system.32