Why does intuitionistic logic require constructive proof?

Intuitionistic logic requires constructive proof because it defines mathematical objects as mental constructions rather than abstract entities existing independently of our ability to demonstrate them. This approach, pioneered by L.E.J. Brouwer, rejects the classical law of excluded middle—the principle that for any proposition $p$, either $p$ or not $p$ must be true.12

In classical logic, one might prove an existence claim by showing that its negation leads to a contradiction. Intuitionists argue this is nonconstructive and mathematically meaningless, particularly regarding infinite sets, because it fails to provide a method to actually construct or identify the object in question. For instance, while one can decide if two natural numbers are equal, it may be impossible to do so for two real numbers, as that would require knowing an infinite number of digits.12

By abandoning principles like the law of excluded middle and double negation, intuitionistic logic ensures that a proof of existence is synonymous with the ability to construct the object, thereby grounding mathematics in verifiable, mental activity rather than potentially empty formal assertions.123