TITLE: Could I be using proof by contradiction too much?
QUESTION [60 upvotes]: Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it directly, thereby reaching a "contradiction." Is this a bad habit?  I don't know why, but there's something incredibly satisfying about proof by contradiction.

REPLY [5 votes]: I think proofs by contradiction are fine; in fact, if one keeps their eyes open for an inconsistency in the foundations of mathematics, it could be a multimillion dollar endeavor. In particular:
STEP 1: Find an inconsistency in ZFC (or other widely used set of axioms).
STEP 2 (Important): Don't tell anyone about it.
STEP 3: Gather a list of the remaining millennium prize problems.
STEP 4: Prove each one by contradiction. By this I mean something like

Assume the Riemann Hypothesis is false. Then ( * cleverly and discretely exhibit inconsistency in ZFC * ), which is a contradiction. Thus RH is true. Please send the check to ( * insert address here * ).

STEP 5: ??????
STEP 6: Profit