Metamathematics

Metamathematics is the mathematics of mathematics itself — the study of formal systems as mathematical objects. The term was coined by David Hilbert to describe the kind of reasoning required for his foundational program: to prove that a formal system (such as arithmetic) is consistent, you must reason about the system from outside it, treating proofs and theorems as combinatorial objects subject to mathematical analysis.

Hilbert believed metamathematical reasoning could be done using only finitary methods — reasoning about concrete symbol strings without appeal to infinite objects. This restriction was essential: if you could prove classical mathematics consistent using methods the intuitionists accepted, you would vindicate classical mathematics against intuitionist critique.

Kurt Gödel's incompleteness theorems were metamathematical results. They treated formal systems as objects of study and proved properties about what such systems can and cannot prove. The second theorem — that no sufficiently strong system can prove its own consistency — showed that finitary metamathematics has limits. To prove a system consistent requires stepping outside to a stronger system, which then requires its own justification. The hierarchy of metamathematical justification has no self-certifying foundation.