Why do intuitionists question the principle that every mathematical statement is either true or false?

Intuitionists, following the work of L.E.J. Brouwer, question the law of excluded middle—the principle that every proposition is either true or false—primarily because they view mathematics as a system of mental constructions rather than a collection of objective, pre-existing truths.12

The core of this objection lies in the treatment of infinite sets. While the principle holds for finite sets where one can inspect every element to verify a property, it fails for infinite sets because there is no exhaustive way to carry out such an inspection. For example, while one can decide if two natural numbers are equal, it may be impossible to determine if two real numbers are equal, as doing so might require knowing an infinite number of decimal digits.2

Consequently, intuitionists argue that nonconstructive arguments—those that assume a statement is true simply because its negation leads to a contradiction—are mathematically meaningless in these contexts. By rejecting this principle, they ensure that mathematical existence is tied to the ability to construct the object in question.12