Why some people don't like proofs by contradiction Possible Duplicate:
  Are the &ldquo;proofs by contradiction&rdquo; weaker than other proofs?  




I have been active on this site for two months and on a few occasions I noticed that some people judge contradiction proofs as being less direct(and less elegant) than proofs which do not use contradiction. 

In my first year of college I gave headaches to my seminar teacher and my colleagues because I used most of the time proof by contradiction. For me it seemed so natural to argue by contradiction whenever I didn't have any idea to how to proceed in solving the problem directly. At least when you prove something by contradiction, you have a start point, a supplementary hypothesis on which you can develop the following arguments searching for a contradiction with the hypothesis or previous work (theorems, problems, etc.).

Therefore, my question is:


  Why do some consider that contradiction proofs are not that good as direct proofs?
I find it harder to read and proofread proofs by contradiction for the following reason: In an ordinary proof, one is trying to show $P \implies Q$. When I read it, I will have in my head a few examples of different things that obey $P$, and I'll check that each step in the proof is consistent with my examples. Hopefully, the last line of the proof will be "$Q$", and everything will work.

In a proof by contradiction, we start out assuming $P \wedge NOT(Q)$. If the theorem is true, there are no examples of things of things which obey $P$ and not $Q$, so I can't think of any examples. 

Note that this problem does not arise in proofs which start out by assuming $NOT(Q)$ and deduce $NOT(P)$ at the end, since I can think of examples of things which obey $NOT(Q)$. I have taken to starting these proofs by writing "We establish the contrapositive, that $NOT(Q)$ implies $NOT(P)$" rather than my standard "This proof proceeds by contradiction. Assume...". My hope is that this will help my readers understand that there are still examples available for them to think about.
To prove something is impossible, you probably can't have a constructive proof and it's ease to understand the counterproof: You just assume that it's possible and derive a contradiction and you're done. Any other way to prove an impossibility?