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.

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### Famous quotes containing the words mathematics and/or classical:

“Why does man freeze to death trying to reach the North Pole? Why does man drive himself to suffer the steam and heat of the Amazon? Why does he stagger his mind with the *mathematics* of the sky? Once the question mark has arisen in the human brain the answer must be found, if it takes a hundred years. A thousand years.”

—Walter Reisch (1903–1963)

“Several *classical* sayings that one likes to repeat had quite a different meaning from the ones later times attributed to them.”

—Johann Wolfgang Von Goethe (1749–1832)