Metamathematics
In general,
metamathematics or
meta-mathematics is reflection 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.
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, is arguably the greatest achievement of metamathematics and the philosophy of mathematics to date.
*
meta-*
consistency*
completeness*
decidable*
mathematical logic*
model theory*
proof theory*
Douglas Hofstadter, 1980.
Gödel, Escher, Bach. Vintage Books. Aimed at laypeople.
*
Stephen Cole Kleene, 1952.
Introduction to Metamathematics. North Holland. Aimed at mathematicians.
*
meta-complexity context
