Classical mathematics
Encyclopedia
In the foundations of mathematics
, classical mathematics refers generally to the mainstream approach to mathematics
, which is based on classical logic
and ZFC set theory. It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics; theories other than those usually used by working mathematicians are thus non-classical mathematics. For example, some approaches to non-standard analysis can also be considered an example of non-classical mathematics. In practice, the most common non-classical systems are used in constructive mathematics.
Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist
and other objections to the logic, set theory, etc., chosen as its foundations, such as have been expressed by L. E. J. Brouwer. Almost all mathematics, however, is done in the classical tradition, or in ways compatible with it.
Defenders of classical mathematics, such as David Hilbert
, have argued that it is easier to work in, and is most fruitful; although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematics could not (or could not so easily) attain, on the whole they argue it is the other way round.
In terms of the philosophy and history of mathematics, the very existence of non-classical mathematics raises the question of the extent to which the foundational mathematical choices humanity has made arise from their "superiority" rather than from, say, expedience-driven concentrations of effort on particular aspects.
Foundations of mathematics
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...
, classical mathematics refers generally to the mainstream approach to mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, which is based on classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
and ZFC set theory. It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics; theories other than those usually used by working mathematicians are thus non-classical mathematics. For example, some approaches to non-standard analysis can also be considered an example of non-classical mathematics. In practice, the most common non-classical systems are used in constructive mathematics.
Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist
Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
and other objections to the logic, set theory, etc., chosen as its foundations, such as have been expressed by L. E. J. Brouwer. Almost all mathematics, however, is done in the classical tradition, or in ways compatible with it.
Defenders of classical mathematics, such as David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, have argued that it is easier to work in, and is most fruitful; although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematics could not (or could not so easily) attain, on the whole they argue it is the other way round.
In terms of the philosophy and history of mathematics, the very existence of non-classical mathematics raises the question of the extent to which the foundational mathematical choices humanity has made arise from their "superiority" rather than from, say, expedience-driven concentrations of effort on particular aspects.
See also
- Constructivism (mathematics)Constructivism (mathematics)In the philosophy of mathematics, constructivism asserts that it is necessary to find a mathematical object to prove that it exists. When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its...
- FinitismFinitismIn the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
- IntuitionismIntuitionismIn the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
- Non-classical analysisNon-classical analysisIn mathematics, non-classical analysis is any system of analysis, other than classical real analysis, and complex, vector, tensor, etc., analysis based upon it.Such systems include:...
- UltrafinitismUltrafinitismIn the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics which are called ultrafinitism...