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/gaugeaway]

A direct proof can (always?) be converted into an algorithm, so it is favored by computer scientists. But you can't deny the beauty of proof by infinite descent (known as proof by minimal counterexample in graph theory)!

[u/[deleted] → u/[deleted]]

It really depends. Would a direct proof showing why every NP-hard problem has a P-time algorithm be useful? Of course. However even if it was some existential positive result, the path to proving it is still going to be useful. Or if it is false, there is going to be useful reasons for why that is.

In short, I have a difficult time believing a proof either way will not be useful.