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Why does Euclid demonstrate several props by their converse or through a reduction to the absurd? In what way is proving that the converse of the prop is absurd preferable to proving constructively what the prop sets out to do.

Tom
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  • Often it’s simply efficient. Lots of Euclids proofs come in pairs. First a constructive proof in a forward direction of a proposition, then proof by contradiction of the backward direction, using the just-demonstrated truth of the forward direction. – RobinSparrow Feb 02 '24 at 04:54
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    Reductio ad absurdum can be convenient because it gives you an extra starting premise to work with, and because it asks that you prove only that something —anything!— goes wrong. In the context of proving a converse, there's already a reasonable candidate for what could go wrong. – Blue Feb 02 '24 at 05:06
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    I wonder what you think the word "converse" means. Do you think it means "negation"? – user14111 Feb 02 '24 at 07:52

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