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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 ...
Analyses or descriptions of an existing theory would be considered meta-theories. [2] If the subject matter of a theoretical statement consists of one or multiple theories, it would also be called a meta-theory. [3] For mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. [4]
Metamath is a formal language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. [2] Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others.
Deduction theorem. In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A → B, it is sufficient to assume A as a hypothesis and then proceed to derive B. Deduction theorems exist for both ...
Metatheorem. In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory. [citation needed] A formal system is determined by a formal ...
Chaitin also writes about philosophy, especially metaphysics and philosophy of mathematics (particularly about epistemological matters in mathematics). In metaphysics, Chaitin claims that algorithmic information theory is the key to solving problems in the field of biology (obtaining a formal definition of 'life', its origin and evolution) and neuroscience (the problem of consciousness and the ...
Metalogic is the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. [1] Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that ...
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities. Major themes that are dealt with in philosophy of mathematics include: Reality: The question is whether mathematics is a pure product of human mind or whether it has some reality by itself.