Metamathematics

https://en.wikipedia.org/wiki/Metamathematics

Metamathematics is the study of mathematics itself using mathematical methods. The study produces metatheories, which are mathematical theories about mathematical theories.

Metamathematics arose due to David Hilbert's attempt to secure the foundations of mathematics, in the early XX century.

Stephen Kleene said in 1952 that metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic".

https://en.wikipedia.org/wiki/Stephen_Cole_Kleene

An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system.

Informally, this means that a proposition like "2+2=4" is classified as belonging to mathematics, while a proposition like "'2+2=4' is valid" belongs to metamathematics.

In this case, mathematics is the object language, while metamathematics is the meta language used to discussed the object language.

Milestones

• non-Euclidean geometries

• discovery of hyperbolic geometry

• Begriffsschrift by Frege

• Principia Mathematica by Russell and Whitehead

• Gödel's incompleteness theorem

• Tarski's definition of model-theoretic satisfaction

• Entscheidungsproblem