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) owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p. 59). An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
👉 Metamathematics in the context of Gregory Chaitin
Gregory John Chaitin (/ˈtʃaɪtɪn/ CHY-tin; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic (Solomonoff–Kolmogorov–Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy.
Metamathematics in the context of Peano arithmetic
In mathematical logic, the Peano axioms (/piˈɑːnoʊ/, [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.
The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic.
View the full Wikipedia page for Peano arithmeticMetamathematics in the context of Meta (prefix)
Meta (from Ancient Greek μετά (metá) 'after, beyond') is an adjective meaning 'more comprehensive' or 'transcending'.
In modern nomenclature, the prefix meta can also serve as a prefix meaning self-referential, as a field of study or endeavor (metatheory: theory about a theory; metamathematics: mathematical theories about mathematics; meta-axiomatics or meta-axiomaticity: axioms about axiomatic systems; metahumor: joking about the ways humor is expressed; etc.).
View the full Wikipedia page for Meta (prefix)Metamathematics in the context of Alfred Tarski
Alfred Tarski (/ˈtɑːrski/; Polish: [ˈtarskʲi]; born Alfred Teitelbaum; January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, type theory, and analytic philosophy.
Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, in 1939 he immigrated to the United States, where in 1945 he became a naturalized citizen. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983.
View the full Wikipedia page for Alfred TarskiMetamathematics in the context of Instance (computer science)
In computer science, an instance or token (from metalogic and metamathematics) is an occurrence of a software element that is based on a type definition. When created, an occurrence is said to have been instantiated, and both the creation process and the result of creation are called instantiation.
View the full Wikipedia page for Instance (computer science)