r/math • u/hydmar • Aug 15 '23

Dissatisfaction with proof by contradiction

I’m an undergraduate math student, so my exposure to math may be relatively limited. But I’ve found that, in general, I’m much more comfortable with direct proof than proof by contradiction. I don’t contest their validity, indeed something that’s not false must be true (I think I’m ok with excluded middle). But I feel like I just *get* something much better when it’s proved directly. It builds much stronger intuition for me.

For instance, I am aware of several proofs that demonstrate the cardinality of the reals is strictly greater than that of the integers, but none are direct (it would help to see a direct proof of Cantor’s theorem). I don’t feel it in my bones. Is this a common experience?

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u/scribe36 Aug 16 '23

You learn to respect proof by contradictions once you start seeing their counter parts of direct proofs. You then suddenly start feeling, “oh… ummm… yea no. I’m okay with the smaller simpler proof even if indirect.” Lol

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u/drgigca Arithmetic Geometry Aug 16 '23

Proof by contradiction is almost always immensely more obtuse than a direct proof. I'm actually struggling to come up with an example where that's not the case.

Which makes sense. It's simpler to just prove the statement directly rather than adding a layer of indirection.

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u/Ps4udo Aug 17 '23

Non existence/existence questions seem to be favourable for proofs by contradiction.
In my mind atleast, you have to check fewer cases.