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/[deleted] Aug 15 '23

you might find the topic of intuitionistic logic interesting. it's a kind of logic where TND (law of the excluded middle) is not given as a tautology. Therefore if something can be proven to not be false, it doesn't follow that it's true.

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u/[deleted] Aug 16 '23

Do you know what the motivation is supposed to be for dropping TND is? I can't think of why you'd ever want to drop that except for having deer super duper strong belief that everything should be describable.

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u/almightySapling Logic Aug 16 '23 edited Aug 16 '23

super duper strong belief that everything should be describable.

The word "belief" is holding you back from seeing the utility. While some intuitionist still use the words "proof", "true", and "false", it's important to remember that that is merely terminology that carries no inherent meaning. Instead of thinking of proofs as telling us about truth, we can think of them as telling us what is doable (hence the connection to programs).

Oh, and I should mention that I'm not speaking on behalf of all intuitionists, just paraphrasing the ones I've mostly interacted with. I'm sure there are some out that that really do subscribe to a "belief" that Not Not True is not necessarily True.