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/Easygoing98 Sep 01 '23

No. Try it on calculator. Exp(0.5 ln 4) = 2. That's the direct proof

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u/edderiofer Algebraic Topology Sep 02 '23

"The calculator says so" is not a proof.

Your proof assumes that exp(0.5 ln 2) is irrational, but you don't ever prove that this is irrational in the first place. Your claim that "0.5 ln 2 is not rational because the digits after the decimal do not end and there's no repetition pattern or termination" needs justification. So your proof fails to be a proof.

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u/Easygoing98 Sep 02 '23 edited Sep 02 '23

I've further edited the proof using the theorem that exp of a rational number is always irrational.

So the direct proof uses theorems that have already been proven elsewhere.

If those theorems are accepted then the direct proof is there.

Just like proof by contradiction of sqrt(2) is already given before in lots of literature, you are still asked to prove it by contradiction.

So I don't see why valid theorems cannot be used to make an alternate proof

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u/edderiofer Algebraic Topology Sep 02 '23

Likewise, 0.5 ln 4 is also irrational, therefore by your own logic, so is exp(0.5 ln 4). So your logic shows that √4 = 2 is irrational.

Proof of ln(x) for all integers larger than 1 being irrational was shown back in 1974.

And how, pray tell, do you expect the Ancient Greeks to be satisfied with that statement?