Metathematic

Metathematic refers to a theoretical approach in mathematics that deals with the properties and structure of mathematics itself. It is a field concerned with the study of mathematical languages, systems, and the very nature of mathematical reasoning. This can involve exploring the foundations of mathematics, its different axiomatic systems, and the logical principles that underpin mathematical proofs.

One key aspect of metathematics is the investigation of consistency, completeness, and decidability within formal mathematical systems. For instance, Gödel's incompleteness theorems are a foundational result in metamathematics, demonstrating inherent limitations in formal axiomatic systems. The field also examines the relationship between different mathematical theories and how they can be translated or compared.

In essence, metathematics takes a step back from doing mathematics to analyze how mathematics is done. It is closely related to mathematical logic, philosophy of mathematics, and theoretical computer science. The goal is to achieve a deeper understanding of what mathematics is, what can be proven within its frameworks, and the underlying principles that make mathematical knowledge possible. It is a meta-level inquiry that seeks to clarify the boundaries and capabilities of mathematical thought.