Metamathematics
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In general, meta-mathematics is knowledge about mathematics seen as an entity/object in the humans consciousness and culture. More precisely, metamathematics is mathematics used to study mathematics. It was originally differentiated from ordinary mathematics in the 19th century to focus on what was then called the foundations problem in mathematics. Important branches include proof theory and model theory. The original meaning of David Hilbert is closest to proof theory (see Hilbert's program).
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.
On the other hand, quasi-empiricism in mathematics, the cognitive science of mathematics, and ethno-cultural studies of mathematics, which focus on scientific method, quasi-empirical methods or other empirical methods used to study mathematics and mathematical practice by which such ideas become accepted, are non-mathematical ways to study mathematics.
See Richard's paradox for an example of the types of contradictions which can easily occur if one doesn't distinguish between mathematics and metamathematics. Some modern thinkers in the field include Bertrand Russell, Paul Benacerraf, Hilary Putnam, Willard Van Orman Quine, Kurt Gödel, Gregory Chaitin, Jules Richard, and Alfred Tarski.
Further reading
- Douglas Hofstadter, 1980. Gödel, Escher, Bach. Vintage Books. For laypersons.
- Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. For mathematicians.