Metamathematics

In general, metamathematics or meta-mathematics is a scientific reflection and knowledge about mathematics seen as an entity/object in human consciousness and culture. More precisely, metamathematics is mathematics used to study mathematics or philosophy of mathematics. Mathematics about mathematics was originally differentiated from ordinary mathematics in the 19th century to focus on what was then called the foundational crisis of mathematics. Richard's paradox is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics.

For example, one of the themes of metamathematics is the analysis of (and hence also discussions about) mathematical elements which are (necessarily) true (or false) in any mathematical system.

Many issues regarding the foundations of mathematics (there is no longer necessarily considered to be any one "problem") and the philosophy of mathematics touch on or use ideas from metamathematics. The working assumption of metamathematics is that mathematical content can be captured in a formal system, usually a first order theory or axiomatic set theory.

Metamathematics is intimately connected to mathematical logic, so that the histories of the two fields largely overlap. Serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift. David Hilbert was the first to invoke the term "metamathematics" with regularity (see Hilbert's program). In his hands, it meant something akin to contemporary proof theory. Another important contemporary branch is model theory. Other leading figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Stephen Kleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, and most important, Alfred Tarski and Kurt Gödel. In particular, Gödel's proof that, given any finite number of axioms for Peano arithmetic, there will be true statements about that arithmetic that cannot be proved from those axioms, a result known as the incompleteness theorem, is arguably the greatest achievement of metamathematics and the philosophy of mathematics to date.