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| ``Re: Nested prooves by contraction?''
by PrimeFan on 2008-07-02 18:11:18 |
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| Is there a context in which this question arose?
Yes, Wkbj79's entry on proof by contradiction. An earlier version of the entry had set off a firestorm of questions regarding what professional mathematicians disapprove of proof by contradiction. Paul Nahin names two in his book on Euler's relation, (he does not count Marylin Vos Savant as such). The current version of the entry reads:
"Proofs by contradiction can become confusing. This is especially the case when such proofs are nested; i.e., a proof by contradiction occurs within a proof by contradiction."
It was that last sentence specifically which made me wonder.
Me personally, I'm not a professional. Amateurs are almost certain to encounter proofs by contradiction very early on (quite likely Euclid's proof that there are infinitely many primes). My first impression was that they were somehow very fair-minded. Pros can be notorious for dismissing things out of hand with great speed (a few cranks make them suspicious of all strangers). A proof by contradiction, in my opinion, shows a willingness to consider a possibility other than the one one is trying to prove. Instead of just dismissing the other, a proof by contradiction, at least a valid one, gives the other a fair shake, not dismissing it until conclusively proving why it's wrong.
Of course an attempted proof by contradiction can fall prey to a subtle mistake that invalidates the whole thing. But so can an attempt at a proof by construction, especially if one does not properly check that a specific example of the construction is indeed valid. |
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