Definition

Noun: The branch of mathematics and logic that studies the logical structure, foundations, and properties of mathematical systems themselves. It involves the analysis of mathematical reasoning, proofs, and theories using logical and mathematical methods.

Usage

Metamathematics is a formal, academic field. It is used as a singular, uncountable noun to refer to the discipline as a whole. * The consistency of the axioms is a central question in metamathematics. * His work lies at the intersection of logic and metamathematics.

Examples

• Gödel's incompleteness theorems are landmark results in metamathematics.
• Researchers use metamathematics to investigate what can and cannot be proven within a given mathematical system.
• The course covers the basics of proof theory, a core area of metamathematics.

Advanced Usage

• Metamathematical analysis: The application of metamathematical techniques to study a specific theory.
  • A metamathematical analysis of arithmetic revealed inherent limitations.
• Metamathematical result: A theorem or finding produced by metamathematical study.
  • The proof of independence was a profound metamathematical result.

Variants and Related Words

• Metamathematical (adjective): Of or relating to metamathematics.
  • The metamathematical implications of the theorem are significant.
• Metatheory: A theory whose subject matter is another theory. Metamathematics is often considered the metatheory of mathematics.

Synonyms

• Foundations of mathematics
• Mathematical logic (in a broad sense, though this is a wider field)
• Proof theory (a major subfield of metamathematics)

Different Meanings

This term is highly specialized and does not have common everyday meanings. Its meaning is consistently tied to the logical study of mathematical systems.

Noun

1. the logical analysis of mathematical reasoning