classical mathematics (changes) in nLab

Showing changes from revision #7 to #8: Added | Removed | Changed

Content

Idea

Classical mathematics is mathematics as it is “normally” practised (or, sometimes, as it used to be practiced), and particularly using commonly accepted foundations.

This is a vague term, but some of the things that it might mean are:

• use of classical logic and the axiom of choice, in contrast to constructive mathematics;

• free use of power sets and infinite sets, in contrast to predicative mathematics and finite mathematics;

• working in Set, in contrast to working internally to some other topos or more general category;

• allowing violating the evil principle of equivalence or otherwise other violating the normative perspectives ofhigher category theory;

• use of non-structural aspects of material set theory.

Despite the name, classical mathematics is only about 100 years old; anything much before that can usually interpreted just as well nonclassically as classically. In fact, it was developments in mathematics around the beginning of the 20th century which sparked the famous “foundational crisis,” leading to the development of intuitionistic, constructive, and predicative mathematics as alternatives, while the approaches accepted by the mainstream were called “classical” in contrast.

Note that the term ‘classical’ also has meanings within many specific fields of mathematics that may have nothing in particular to do with ‘classical mathematics’ as a whole. Compare also ordinary mathematics.

• intuitionistic mathematics

• constructive mathematics

• structural set theory, ETCS

• type theory, homotopy type theory