Intuitionists

Intuitionism is a school of thought in the philosophy of mathematics and logic that emphasizes the role of mental constructions in establishing mathematical truths. Founded by L.E.J. Brouwer in the early 20th century, intuitionism holds that mathematical objects and statements are meaningful only if they can be constructed or proven by a human mind. This contrasts with other mathematical philosophies that allow for the existence of mathematical entities independently of human comprehension.

A key principle of intuitionism is the rejection of the law of excluded middle, which states that for any proposition P, either P is true or its negation is true. Intuitionists argue that this law can only be applied when there is a constructive method for determining the truth value of P or its negation. Consequently, they do not accept proofs by contradiction for statements whose existence is not constructively established.

In intuitionistic logic, a proof of existence must provide a method for constructing the object in question, and a proof of a disjunction (P or Q) must provide a method for proving either P or Q. This constructive approach leads to a different set of accepted theorems compared to classical mathematics. For instance, the completeness of the real numbers is accepted in classical mathematics but requires a nuanced understanding or modification within an intuitionistic framework.

Intuitionism has had a significant impact on the development of constructive mathematics and has influenced fields such as computer science, particularly in areas related to computability and proof theory. While not universally adopted, its focus on constructive proof remains an important perspective in the foundations of mathematics.