Constructive mathematics

Intuitionism, as the crucial philosophy of mathematics, holds that math, as a human-driven endeavor, should not pursue some absolute platonic truths, but rather concern itself with what we, humans, can construct.

This has introduced the constructive mathematics approach, in which all math objects must be constructible. This stance holds that a proof is not complete unless a constructible instance of that proof is provided (a "witness" to the proof).

One of the major impacts of this requirement is the rejection of the law of the excluded middle, which is one of the cornerstones of the classical mathematical reasoning.

In classical math school, the fundamental method of deduction, proof by contradiction, depends on the law of excluded middle; instead of directly proving that a statement is true, we can prove that the opposite is false. For example, instead of proving the existence of an object, we can try assuming the opposite; then, if during the course of the proof, we encounter a contradiction, it doesn't only render the assumption false, but at the same time, it shows that the opposite is true (that an object does exist).

This practice is disallowed in constructive mathematics, of which intuitionistic logic is the most well-known representative, where such proofs are deemed pointless, since they fail to construct a witnessing instance.

Although it may seem counter-intuitive, intuitionism holds that "not false" is not necessarily "true", and vice versa. Also, the existence of a third alternative to provably true or provably false is the direct implication of Gödel's first incompleteness theorem, which states that some theorems are not provable at all.