Classical mathematics, as a term of art in mathematical logic, refers generally to mathematics constructed and proved on the basis of classical logic and ZFC set theory, i. e. using Zermelo-Fraenkel set theory and the axiom of choice. Non-classical mathematics, in general, refers to mathematics constructed on a non-classical system of logic, and/or some non-classical set theory.

Classical mathematics therefore refers to making the foundational decisions for mathematics in agreement with the mainstream of modern mathematics as of 2004; non-classical mathematics involves making some alternative choices. Thus, for example, non-standard analysis can also be considered an example of non-classical 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. Almost all mathematics, however, is done in the classical tradition, or compatibly with it.

Defenders of classical mathematics argue that it is easier to work in, and is most fruitful; although they acknowledge non_classical mathematics has at time lead 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. Opponents of classical mathematics will argue that of course classical mathematics will be more fruitful if the vast majority of mathematical energy is expended on it.

In terms of the philosophy and history of mathematics, the very existence of non_classical mathematics raises the question of to what extent the foundational mathematical choices we have made as a society, are due to them being "superior", or is it purely due to contingencies of where we have chosen to expend the majority of our mathematical energy?

See also non-classical analysis.