TITLE: Why some people don't like proofs by contradiction
QUESTION [7 upvotes]: Possible Duplicate:
Are the “proofs by contradiction” 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?

REPLY [10 votes]: Philosophical issues aside, a point that's been left out: what's better when proving need not be what's better when writing down the proof.
You say "For me it seemed so natural to argue by contradiction…", and you're right: yes, it is often easier when proving to argue by contradiction, but the version without contradiction is often easier to read (and therefore better to write).
When you're trying to prove $P \implies Q$, if you prove by contradiction, then you get to assume both $P$ and $\lnot Q$ before exploring their consequences, so in principle it's never worse than assuming $P$ alone: proving by contradiction never hurts. But after you've finished the proof, it's worth going over your proof and seeing if it can be written directly. 
That is, the issue (stylistically speaking) is not so much with proof by contradiction per se, but with writing a proof in terms of contradiction when it's not necessary or helpful. ("Proof by unnecessary contradiction".) See e.g. Mathematical Writing (by Donald Knuth, Tracy Larrabee, and Paul M. Roberts) for this (emphasis added):

The proof above actually commits another sin against mathematical exposition, namely the unnecessary use of proof by contradiction. It would have been better to use a direct proof…

Sometimes (often) the direct proof may be easier to read and more illuminating, and sometimes it may not. It is always worth considering both versions and choosing the one that's easier to read. You owe it to your readers to think carefully about how best to organise your proof.