How did L.E.J. Brouwer and Arend Heyting develop intuitionism?

Intuitionism was introduced by the Dutch mathematician L.E.J. Brouwer in the early 20th century as a response to the need for a secure foundation of mathematics. Brouwer contended that mathematical objects are mental constructions and argued that many classical mathematical principles were nonconstructive and therefore meaningless.123

Brouwer’s fundamental insight was that nonconstructive arguments could be avoided by rejecting the law of excluded middle—the principle that for any proposition, either it or its negation must be true. He argued this principle may fail when dealing with infinite sets, where a definitive proof for a property may not be possible through finite inspection.3

While Brouwer viewed mathematics as language-independent, his student Arend Heyting further developed the school by formalizing its principles. Heyting established a formal language for first-order intuitionistic arithmetic, providing a structured framework for the intuitionistic rejection of nonconstructive proofs and the double-negation axiom. Together, their work shifted the focus of mathematical assertion toward the requirement of explicit proof.23