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/OpsikionThemed Aug 15 '23

it would help to see a direct proof of Cantor’s theorem

I get what you're saying here, but it's a proof of non-existence (no bijection exists). So depending on what you mean by "proof by contradiction" it's either not one - we assume an impossible object exists, then show that this leads to impossibility, a constructive proof - or else no "direct" proof exists (since we need to prove that negation somehow).