# 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.
