Classical Mathematics

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. 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.

Other articles related to "classical mathematics, mathematics, classical":

Criticism Of Non-standard Analysis - Bishop's Criticism
... In the view of Errett Bishop, classical mathematics, which includes Robinson's approach to nonstandard analysis, was non-constructive and therefore deficient in numerical meaning (Feferman 2000) ... analysis in teaching as he discussed in his essay "Crisis in mathematics" (Bishop 1975) ... formalist program he wrote A more recent attempt at mathematics by formal finesse is non-standard analysis ...
Paul Dirac - Career
... this mathematical form had the same structure as the Poisson Brackets that occur in the classical dynamics of particle motion ... an analogy between the Poisson brackets of classical mechanics and the recently proposed quantization rules in Werner Heisenberg's matrix formulation of quantum mechanics ... to a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin ħ/2, itself subject to quantization conditions involving anti-commutators ...
Second-order Arithmetic
... a foundation for much, but not all, of mathematics ... can prove essentially all of the results of classical mathematics expressible in its language ... Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying ...

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