Proof by contradiction only works if you are certain that you are working with a scenario that is a boolean. I can imagine a scenario where someone thinks "true" or "false" are the only answers, only to draw a conclusion based on proof by contradiction, when in reality, they are ignoring another possible answer. I do not know how this would play out in practice... maybe in quantum mechanics? I'm not sure if that is even relevant to this. Just a thought...
I would like to hear more of the logic behind the following quote:
"[I]f a proof can be done without contradiction, then that is usually a better option, because you never enter an "imaginary" world where you assume something you are hoping to show is false."
It seems to me that a proof is a proof, right? I don't understand why mathematicians have aversions to certain types of proofs. I have been told that proofs by induction are considered ineloquent and thus should be avoided when possible. Why are some proofs superior to others? "All proofs are equal, but some proofs are more equal than others." :-)
Yours truly,
Batman
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