TITLE: Aren't constructive math proofs more "sound"?
QUESTION [17 upvotes]: Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs?  How do we know the "regular" kind of mathematics are free of paradox without a proof construction?

REPLY [7 votes]: "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether."
-David Hilbert