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| ``Re: Reject proof by contradiction?''
by Algeboy on 2006-06-27 12:20:06 |
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| I have to generally agree with silverfish that there is nothing wrong with rejecting "proof by contradiction" BUT only if you are prepared to reject the prinicipal of the excluded middle (PEM): ie that a well-formed statement is either true or false, and nothing in between. People who feel qweeze about proof by contradiction don't seem skeptical about PEM in my experience, but I had a logician once show me PEM implies the validity of proof by contradiction. So if you have PEM as an axiom logic, you have proof by contradiction.
But I really think rmilson has the best point on this: proof is not about being correct, it is about convincing someone else that you are correct. And to the extent that proof by contradiction confusses people, then it is a weaker proof. So a mathematician is well within his rights, and perhaps prudent, to reject one proof technique over another, if he/she feels this better explains it to a someone.
I did once discuss with a colleague the idea if removing PEM as an axiom would make Godel's incompleteness theorem fail, (these proofs are carefully balanced on PEM) or would an alternate version of the problem appear. The real problem was, with middle ground between true and false, nothing made sense! So live in the world we've made, pitfalls and all. |
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