With good reason, we often prefer direct proofs over proofs by contradiction. Direct proofs often carry information about how to construct the mathematical objects whose existence is being asserted. But more importantly, direct proofs often paint a fuller picture of mathematical reality. When one proves an implication p → q directly, one assumes p and then derives various further consequences p1, p2, and so on, before ultimately concluding q. Thus, one has derived a whole context about what it is like in the p worlds.
