Constructivism (mathematics)

In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a mathematical object to prove that it exists.

Description

When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence, according to constructivism. This viewpoint involves a verificational interpretation of the existence quantifier, which is at odds with its classical interpretation.

There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of David Hilbert and Bernays, the constructive recursive mathematics of Nikolai Aleksandrovich Shanin and Andrey Markov, and Bishop's program of constructive analysis.

Constructivism also includes the study of constructive set theories such as IZF and the study of topos theory.

Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.

Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

See also

• Constructive logic
• Constructive type theory
• Constructive analysis
• Constructive non-standard analysis
• Computability theory
• Constructive proof
• Finitism
• Game semantics
• Intuitionism
• Intuitionistic type theory

External links

• Constructivism (mathematics) @ Wikipedia.org