Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) is due to David Hilbert's attempt of proving the consistency of mathematical theories by proving a proposition about a theory itself, i.e. specifically about all possible proofs of theorems in the theory; in particular, both a proposition A and its negation not A should not be theorems (Kleene 1952, p. 55). However, metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic, among which the consistency problem is only one" (Kleene 1952, p. 59).
Metamathematical metatheorems 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 (Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions that can easily occur if one fails to distinguish between mathematics and metamathematics. Something similar can be said around the well-known Russell's paradox (Does the set of all those sets that do not contain themselves contain itself?).
Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics.
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, in which finitary methods are used to study various axiomatized mathematical theorems (Kleene 1952, p. 55).
Other prominent figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Stephen Kleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, Alfred Tarski and Kurt Gödel. In particular, arguably the greatest achievement of metamathematics and the philosophy of mathematics to date is Gödel's incompleteness theorem: 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.
- Principia Mathematica (Whitehead and Russell 1925)
- Gödel's completeness theorem, 1930
- Gödel's incompleteness theorem, 1931
- Tarski's definition of model-theoretic satisfaction, now called the T-schema
- The proof of the impossibility of the Entscheidungsproblem, obtained independently in 1936–1937 by Church and Turing.
- W. J. Blok and Don Pigozzi, "Alfred Tarski's Work on General Metamathematics", The Journal of Symbolic Logic, v. 53, No. 1 (Mar., 1988), pp. 36–50.
- I. J. Good. "A Note on Richard's Paradox". Mind, New Series, Vol. 75, No. 299 (Jul., 1966), p. 431. JStor
- Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage Books. Aimed at laypeople.
- Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. Aimed at mathematicians.
- Jules Richard, Les Principes des Mathématiques et le Problème des Ensembles, Revue Générale des Sciences Pures et Appliquées (1905); translated in Heijenoort J. van (ed.), Source Book in Mathematical Logic 1879-1931 (Cambridge, Mass., 1964).
- Alfred North Whitehead, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.