When I once realised I don't really understand how and why proof by contradiction works, I started reading about it. And apparently I wasn't the only one who felt there's something wrong about it - constructivists and intuitionists do not accept the law of excluded middle. This is what shocked me the most - I thought logic is something that has been already settled a long time ago. It appears the truth is very different though. Logic is an attempt to describe and systematize the way we reason. And if some people accept the law of the excluded middle, and some don't, then it feels the same as if some people believed 1+2=3, and others 1+2=4. If we can all agree on how basic arithmetic works, why don't we have consensus in more complicated areas of logic?
Human thought is not based on logic, but logic is based on human thought. Do you believe we will eventually have a consensus in this area? People try to solve much more complex problems, such as P vs NP. But shouldn't we focus on the basic laws of reasoning first, especially if we have doubts in that field?
Mathematics exists because of the assumption we think in the same way. So it has to be the case, maths can't be based on our personal opinions or feelings. The existence of constructivism makes it look that way though. I'm writing these words assuming you will understand them the same way I do. Otherwise, What would be the point in writing this?