metamathematical formalism

Formalism is a school of thought in the philosophy of mathematics, primarily associated with mathematician David Hilbert. It posits that mathematics can be reduced to the manipulation of formulas according to set rules, without needing to consider the meaning of these formulas. Formalists believe that the symbols themselves, rather than any ascribed meaning, are the fundamental elements of mathematical thought.12

Metamathematics is a field concerned with the formal structure and properties of mathematical systems, such as the consistency and completeness of axioms. Formalism emerged in the early 20th century as part of a broader effort to establish a secure foundation for mathematics, aiming to demonstrate its reliability and certainty. While Hilbert's program to prove the consistency of mathematics within its own formal system was ultimately shown to be unattainable by Kurt Gödel's incompleteness theorems, formalism remains influential, with many mathematicians implicitly adopting its principles.3456