metamathematics
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Metamathematics is
mathematics used to study mathematics or philosophy of mathematics. Mathematical theorems about mathematics itself were originally differentiated from ordinary mathematical theorems 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
MetamathematikMetamatematicaMetamatematikaMetamatematikkMetamatematykaМетаматематикаMetamatematika元数学
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