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.
@loke I'm not a mathematicians, but I will underline two things.
First, in a formal proof you had to prove that all the initial conditions C1..Cn can be respected. Often this is implicit. Then you introduce the assumption P that you want to disprove. So, if P causes a contradiction, but C1...Cn can be respected, it is P that is impossible to respect, given C1...Cn. Usually C1...Cn are not strong conditions, but rather generic conditions. So often there is no explicit proof about the fact that they can be respected. Often it is true by construction.
Second: not all "proof by contradiction" are strictly "by contradiction", but many of them are simpler cases of "refutation by contradiction" and they are accepted also in intuitionistic logic.
A "refutation by contradiction" is based on this: C1...Cn are initial conditions that can be respected; I suspect that P is not respectable, i.e. that "not P" is true; I assume P and I show that P cannot be respected.
The integer between 0 and 1 is an example of "refutation by contradiction". I suspect that there is no such number; I assume it exists; I obtain a contradiction.
Instead a "proof by contradiction" using the rule of the excluded middle, is less direct, because we assume that if "not P" introduces a contradiction, then "P" must be true, but in the proof we have no directly proved the existence of P. These proofs are not accepted by intuitionistic logic.
