Full Idea
In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S.
Gist of Idea
Showing a disproof is impossible is not a proof, so don't eliminate double negation
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3)
Book Reference
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.7
A Reaction
I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly.