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 Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...

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 a ...

, 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.

References

Mathematical logic {{mathlogic-stub

See also

References