I came across the following in Royden’s Real Analysis:

All students are enjoined in the strongest possible terms to eschew proofs by contradiction!  There are two reasons for this prohibition : First, such proofs are very often fallacious, the contradiction on the final page arising from an erroneous deduction on an earlier page… Second, even when correct, such a proof gives little insight into the connection between A and B, whereas both the direct proof and the proof by contraposition construct a chain of argument connecting A with B.  One reason that mistakes are so much more likely in proofs by contradiction than in direct proofs or proofs by contraposition is that in a direct proof (assuming the hypothesis is not always false) all deductions from the hypothesis are true in those cases where the hypothesis holds, and similarly for proofs by contraposition (if the conclusion is not always true), the deductions from the negation of the conclusion are true in those cases where the conclusion is false.  Either way, one is dealing with true statements, and one’s intuition and knowledge about what is true help to keep one from making erroneous statements.  In proofs by contradiction, however, you are (assuming the theorem is true) in the unreal world where any statement can be derived, and so the falsity of a statement is no indication of an erroneous deduction.

This impassioned attack reminds me of Ionesco’s The Lesson or Stoppard’s Jumpers.  It’s tempting to psychoanalyze Royden and say that he must have an irrational fear of an unreal “make-believe” world or an overwhelming need to have certainty and truth.  But that’s just silly, isn’t it?

1. “In proofs by contradiction, however, you are (assuming the theorem is true) in the unreal world where any statement can be derived, and so the falsity of a statement is no indication of an erroneous deduction.”

But, but… isn’t proving that you are in the “unreal world” the whole point of proofs by contradiction? I think I’m missing his point.

PS — This auto preview-comments-while-you-type feature is pretty snazzy.

2. Ari Nieh says:

There’s nothing particularly wrong with his argument, but I’ve never seen an example of what he describes.

The only problem I’ve seen with proof by contradiction is that novice students will sometimes cite it incorrectly when their proof is really contraposition.

3. I loved his argument! In some sense, proof by contradiction is aligned with nihilism… Plus, I do really think that anytime I contradict myself, I am not sure I am revealing anything interesting about me… Although it might be that this is the whole point.

Who knows… I once heard about a book called “Probability by counterexamples”…

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