Metamathematics

Metamathematics is mathematics used to study mathematics.

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It was originally differentiated from ordinary mathematics in the 19th century to focus on what was then called the foundations problem in mathematics.

Related Topics:
19th century - Foundations problem in mathematics

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Important branches include proof theory and model theory. The original meaning of David Hilbert is closest to proof theory (see Hilbert's program).

Related Topics:
Proof theory - Model theory - David Hilbert - Hilbert's program

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These would nowadays be considered part of Mathematical logic, which (according to Googlefight) is much more widely used than the obsolete term metamathematics & Gödel, Escher, Bach].

Related Topics:
Mathematical logic - Googlefight - Library of Congress - Gödel, Escher, Bach

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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.

Related Topics:
Foundations of mathematics - Philosophy of mathematics - Formal system

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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.

Related Topics:
Quasi-empiricism in mathematics - Cognitive science of mathematics - Ethno-cultural studies of mathematics - Scientific method - Quasi-empirical methods - Empirical methods - Mathematical practice

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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, and Jules Richard.

Related Topics:
Richard's paradox - Bertrand Russell - Paul Benacerraf - Hilary Putnam - Willard Van Orman Quine - Kurt Gödel - Jules Richard

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