Metamathematics

Part of speech: noun

Definitions

1. The branch of logic and philosophy that examines the properties, consistency, and foundations of mathematics itself as a formal system
2. The mathematical study of mathematics, investigating the rules, axioms, and logical structure underlying mathematical theories and proofs
3. A discipline that analyzes mathematical systems by treating mathematics as an object of investigation, examining what can be proven within formal frameworks

Etymology: "Metamathematics" is a compound built from two Greek-derived parts: "meta-", from Ancient Greek "metá" meaning "after" or "beyond" (and in later usage "about" or "on a higher level"), and "mathematics," from Greek "mathēmatikḗ," meaning the study of learning or things learned. Put together, the name describes the study of mathematics itself, reasoning about mathematical systems from the outside rather than working within them. The word is a modern technical coinage rather than something ancient mathematicians would have used. The prefix "meta-" had by then become a productive way to label a discipline that examines another discipline, much as "metaphysics" was used for a topic that followed physics in an old arrangement of Aristotle's works. Here the "meta-" relation is different: the subject studied is mathematics, and the tools are mathematical too, turned back on proofs, formal languages, and consistency.

Synonyms: proof theory