# Dictionary Definition

metamathematics n : the logical analysis of
mathematical reasoning

# Extensive Definition

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 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.

## See also

## References

- Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage Books. Aimed at laypeople.
- Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. Aimed at mathematicians.

metamathematics in German: Metamathematik

metamathematics in Italian: Metamatematica

metamathematics in Hungarian:
Metamatematika

metamathematics in Norwegian:
Metamatematikk

metamathematics in Polish: Metamatematyka

metamathematics in Slovak: Metamatematika

metamathematics in Chinese:
元数学