Section 13 of Kleene's Intoduction to Metamathematics introduces briefly Brouwer's informal intuitionistic school of thought. There he writes that the interpretation of not A is meant to be taken as A implies a contradiction.
From this definition, is it wrong to also interpret not A as there does not exist a proof of A?
If this were the case I would better understand the non-acceptance of double negation elimination. Because then, not not A asserts that there doesn't exist a disproof of A however, from only that assertion we cannot construct an existential proof of A.