TITLE: Are the "proofs by contradiction" weaker than other proofs?
QUESTION [117 upvotes]: I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive.  Could you explain the reason? Do logicians think that proofs by contradiction are somewhat weaker than direct proofs?
Is there any reason that one would still continue looking for a direct proof of some theorem, although a proof by contradiction has already been found? I don't mean improvements in terms of elegance or exposition, I am asking about logical reasons.  For example, in the case of the "axiom of choice", there is obviously reason to look for a proof that does not use the axiom of choice.  Is there a similar case for proofs by contradiction?

REPLY [11 votes]: In order to prove A, let's assume not A.
[Insert 10-page argument here.]
Which of the assertions proved in the foregoing 10 pages are false because they were deduced from the (now proved false) assumption that not A?  Which are true but cannot be considered to have been validly proved because the proofs relied on the false assumption that not A?  And which were validly proved since their proofs did not rely on that assumption?  It can be hard to tell.  And if you saw an assertion proved along the way, you might think it's known to be true.
In that way, a proof by contradiction can be at best confusing.