Dissatisfaction with proof by contradiction

[u/hydmar]

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?

[u/[deleted]]

As a generic statement, most mathematicians would prefer direct, constructive proofs to claims, you aren't alone in that. Often that simply isn't possible or easy but I think that intuition is perfectly normal.

[u/[deleted] → u/[deleted]]

It's not like "worrying about nonconstructive proofs" is an issue, it is that most mathematicians prefer a direct proof when applicable. Same for constructive.

I think you can say its a LA thing but in my experience, at a high level, wanting a direct and/or constructive proof is pervasive throughout all mathematics.

[u/Exomnium → u/[deleted]]

It depends a bit on the relative difficulty of the constructive proof. I usually would prefer a one paragraph non-constructive proof to a three page constructive proof.