Why can proofs in an uninterpreted formal system be studied without considering meaning?

In an uninterpreted formal system, proofs can be studied without considering meaning because the system is defined entirely by its syntactic structure rather than its semantic content.12

A formal system consists of a set of primitive symbols, rules for forming well-formed formulas, and rules of inference that allow for the mechanical derivation of theorems from axioms. Because these components are specified purely syntactically, the validity of a proof depends solely on whether the sequence of formulas follows the established rules of the system, not on what the symbols or formulas signify.3452

Studying these systems as uninterpreted logical calculi allows logicians to analyze the deductive structure of thought in the abstract. This approach offers several advantages:4
* Generality: An uninterpreted system can be applied to various interpretations; by studying the system without meaning, one explores the structure common to many different interpreted systems.2
* Mechanical Verification: Because the rules are syntactic, the process of checking a proof can be performed mechanically, independent of any intuitive understanding of the subject matter.35
* Metamathematical Analysis: This focus on syntax enables the study of properties like consistency, which can be investigated by examining whether an inconsistency can be derived through formal rules alone.1