Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?
The foundational philosophy of Platonist mathematical realism, as exemplified by ...
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Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory. The search for foundations of mathematics is also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?
The foundational philosophy of Platonist mathematical realism, as exemplified by mathematician Kurt Gödel, proposes the existence of a world of mathematical objects independent of humans; the truths about these objects are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world?
The foundational philosophy of formalism, as exemplified by David Hilbert, is based on axiomatic set...
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