How does intuitionism interpret the denial of a mathematical claim?

In the philosophy of mathematics, intuitionism defines mathematical objects as mental constructions rather than objective truths existing independently of the mind. Consequently, the intuitionist interpretation of the denial of a claim differs significantly from classical logic.1

For intuitionists, to assert that a mathematical claim is denied—or to prove its negation—requires a constructive demonstration that the claim leads to a contradiction. Because intuitionism rejects the law of excluded middle (the principle that for any proposition $p$, either $p$ or not $p$ must be true), it does not accept that a statement is true simply because its negation has been proven false.2

This is particularly relevant regarding infinite sets. While one can often decide between $x=y$ and $x\ne y$ for finite cases, intuitionists argue that for infinite sets, one cannot assume a claim is either true or false without a constructive method to verify it. Thus, the double negation principle ($\neg \neg p\supset p$) is not accepted as a universal axiom, as the absence of a contradiction does not equate to a positive construction of the claim.2