I have a question for mathematicians out there.
I understand how a proof by contradiction works. I've seen a lot of them and they often quite ready to follow.
Now, I was watching a maths video going through a proof why there is no integer between 0 and 1, and it was a proof by contradiction.
That's when I realised why they feel uncomfortable to me. I would never attempt to prove something in this way myself, because it feels like any time mistake in my reasoning will lead me to believe that there is a contradiction, and thus make me assume that I have a proof.
It feels as though this is different from a regular proof where a small mistake will lead me to believe that the thing I want to prove is false. Or at least that the approach doesn't work.
In other words it is my impression that a proof by contradiction "fails true" as compared to a regular proof which "fails false".
Now for my question to mathematicians: is my intuition wrong here? Is there a greater risk of getting something wrong with proofs by contradiction rather than regular proofs?
Are there examples of such proofs that have been shown to be incorrect due to a different assumption being false rather than the one that was assumed?
I hope my question makes sense.
I'm not sure I understand what you mean exactly, but I do feel that there's really no such thing as a tiny mistake in a mathematical proof. Any mistake can completely trip you up, whether it's a proof by contradiction or not.
I get that in real world situations and engineering, "good enough" is a thing. But in a mathematical proof, it ... just isn't.
@isaackuo yes. I figured it must be something like that. I was just wondering if it's easier to be tricked into thinking you did everything right when you're working on a proof by contradiction rather than a regular proof?
You use proofs by contradiction implicitly quite often (e.g. "either p or not p" requires the use of proof by contraduction to prove). The branch of logics that recommend what can be proven without p.b.c. doesn't match the intuition you're going for: it excludes everything that requires p.b.c. somewhere internally.
I think you're going for the notion of not being able to imagine what since intermediate statements in the proof mean for simple examples. That is generally harder e.g. for proofs to that try to show that something is impossible, regardless of whether they involve p.b.c. in any way.
