In the
foundations of mathematics, classical mathematics refers generally to the mainstream approach to
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, 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, they argue that on the whole, it is the other way round.
See also
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Constructivism (mathematics)
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Finitism
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Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
*
Non-classical analysis
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Traditional mathematics
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Ultrafinitism
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Philosophy of Mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
References
Mathematical logic
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