 Metamathematics

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 wellknown 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.
Contents
History
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.
Milestones
 Principia Mathematica (Whitehead and Russell 1925)
 Gödel's completeness theorem, 1930
 Gödel's incompleteness theorem, 1931
 Tarski's definition of modeltheoretic satisfaction, now called the Tschema
 The proof of the impossibility of the Entscheidungsproblem, obtained independently in 1936–1937 by Church and Turing.
See also
References
 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 18791931 (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.
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
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 date19010114 birth place = Warsaw, Poland (under Russian rule at the time) death date = death date19831026 death place = Berkeley, California fields = Mathematics, logic,… … Wikipedia