Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. 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 which 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?).

The term "metamathematics" is sometimes used as a synonym for certain elementary parts of formal logic, including propositional logic and predicate logic.



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.

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, in which finitary methods are used to study various axiomatized mathematical theorems.

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.


See also


  • 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.

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  • metamathematics — 1890, from META (Cf. meta ) + MATHEMATICS (Cf. mathematics) …   Etymology dictionary

  • metamathematics — [met΄ə math΄ə mat′iks] n. the logical study of the nature and validity of mathematical reasoning and proof …   English World dictionary

  • metamathematics — noun plural but usually singular in construction Date: circa 1890 a field of study concerned with the formal structure and properties (as the consistency and completeness of axioms) of mathematical systems • metamathematical adjective …   New Collegiate Dictionary

  • metamathematics — metamathematical, adj. metamathematician /met euh math euh meuh tish euhn/, n. /met euh math euh mat iks/, n. (used with a sing. v.) the logical analysis of the fundamental concepts of mathematics, as number, function, etc. [1885 90; META +… …   Universalium

  • metamathematics — noun A branch of mathematics dealing with mathematical systems and their nature …   Wiktionary

  • metamathematics — The theory of formal languages powerful enough to serve as the language of mathematics. In a formal metamathematical treatment, the formulae that occur in mathematics: axioms, theorems, and proofs, are treated as themselves mathematical objects,… …   Philosophy dictionary

  • metamathematics — met•a•math•e•mat•ics [[t]ˌmɛt əˌmæθ əˈmæt ɪks[/t]] n. (used with a sing. v.) math. the study of fundamental concepts of mathematics, as number and function • Etymology: 1885–90 met a•math e•mat′i•cal, adj …   From formal English to slang

  • metamathematics — noun the logical analysis of mathematical reasoning • Topics: ↑mathematics, ↑math, ↑maths • Hypernyms: ↑pure mathematics …   Useful english dictionary

  • metamathematical — metamathematics …   Philosophy dictionary

  • Alfred Tarski — Infobox scientist name = Alfred Tarski caption = birth date = birth date|1901|01|14 birth place = Warsaw, Poland (under Russian rule at the time) death date = death date|1983|10|26 death place = Berkeley, California fields = Mathematics, logic,… …   Wikipedia