Editing Metamathematics (section)

== History ==
Metamathematical [[metatheorem]]s about mathematics itself were originally differentiated from ordinary [[mathematical theorem]]s 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]], [[category theory]], [[recursion theory]] and pure [[model theory]].

Serious metamathematical reflection began with the work of [[Gottlob Frege]], especially his ''[[Begriffsschrift]]'', published in 1879.

[[David Hilbert]] was the first to invoke the term "metamathematics" with regularity (see [[Hilbert's program]]), in the early 20th century. 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.&nbsp;55).

Other prominent figures in the field include [[Bertrand Russell]], [[Thoralf Skolem]], [[Emil Post]], [[Alonzo Church]], [[Alan Turing]], [[Stephen Kleene]], [[Willard Quine]], [[Paul Benacerraf]], [[Hilary Putnam]], [[Gregory Chaitin]], [[Alfred Tarski]], [[Paul Cohen]] and [[Kurt Gödel]].

Today, [[metalogic]] and metamathematics broadly overlap, and both have been substantially subsumed by mathematical logic in academia.