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| ``Re: Nested prooves by contraction?''
by nkirby on 2008-07-02 22:53:52 |
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| Oh I see now.
I basically agree with you. But I'd add that one pitfall is a feeling that proving "If A, then B" is easier using a contradiction argument, because you get to assume more A _and_ not B (which somehow sounds more fun), than a direct argument where you only get to assume A (or not B if you wish to prove the contrapositive). All of this is utterly separate from the philosophical question that led to the firestorm, in which I have no pony since I'm the type to see math as a formal system so that axioms can either be taken or left based on personal preferences and pragmatic goals.
I've heard it said that proofs by contradiction are usually draft proofs, because of exactly this freedom found in assuming more, but that in the polishing process one hopes to tighten the idea of the proof to the point that you can show the nuts and bolts of "why" there was a contradiction, i.e. find a direct argument. |
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