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. In philosophy, an object is a thing, an entity, or a being. ... Trinomial name Homo sapiens sapiens Linnaeus, 1758 Humans, or human beings, are bipedal primates belonging to the mammalian species Homo sapiens (Latin for wise man or knowing man) under the family Hominidae (the great apes). ... Consciousness is a quality of the mind generally regarded to comprise qualities such as subjectivity, self-awareness, sentience, sapience, and the ability to perceive the relationship between oneself and ones environment. ... The word culture, from the Latin colo, -ere, with its root meaning to cultivate, generally refers to patterns of human activity and the symbolic structures that give such activity significance. ... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Richards paradox is a fallacious paradox of mathematical mapping first described by the French mathematician Jules Richard in 1905. ...

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. Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... In mathematical logic, a first-order theory is given by a set of axioms in some language. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...

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. Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... Begriffsschrift is the title of a short book on logic by Gottlob Frege, published in 1879, and is also the name of the formal system set out in that book. ... David Hilbert David Hilbert (January 23, 1862, Wehlau, East Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Hilberts Program was to formalize all existing theories to finite real complete set of axioms, and provide a proof that these axioms were consistent. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, and mathematician, working mostly in the 20th century. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... Emil Leon Post (February 11, 1897 - April 21, 1954) was a Polish-American mathematician and logician. ... Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... Paul Benacerraf is an American philosopher of mathematics based at Princeton University. ... Hilary Whitehall Putnam (born July 31, 1926) is a key figure in the philosophy of mind during the 20th century. ... Gregory J. Chaitin (born 1947) is an Argentine-American mathematician and computer scientist. ... Alfred Tarski (January 14, 1901, Warsaw Poland – October 26, 1983, Berkeley California) was a logician and mathematician of considerable philosophical importance. ... Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...

See also

• meta-
• consistency
• completeness
• decidable
• mathematical logic
• model theory
• proof theory

In epistemology, the prefix meta- is used to mean about (its own category). ... Consistency has several technical meanings: In NASCAR Racing, consistency is a term coined by NASCAR drivers about the frequency of finishing well in the top ten or top five each race as it helps to get enough points to make the Chase For The Cup and win the Nextel Cup... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... The word decidable has formal meaning in computability theory, the theory of formal languages, and mathematical logic. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...

References

• Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage Books. Aimed at laypeople.
• Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. Aimed at mathematicians.

Douglas Richard Hofstadter (born February 15, 1945) is an American academic. ... GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ... Stephen Cole Kleene (January 5, 1909 – January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ...

External link

• meta-complexity context