When we prove a statement, we show it is true. Since contradictions (statements such as "P and not-P.") are never true, we can't ever prove a contradiction. But that's precisely what we do in a proof by contradiction - we show a contradiction to be true, before declaring it absurd. This must mean we are doing something wrong. It must mean that we can't even assume a false statement to begin with. This makes sense because when we assume a statement, we pretend that it's true, but we can't pretend that a false statement is true. It's a logical impossibility. That would be like saying "1 + 1 = 4" is true. Does this mean the "proof by contradiction" method is flawed?
In other words, to prove proposition P, we assume not-P and show this leads to a contradiction. But if P is true to begin with (as we want to prove it), and therefore not-P false, how can we even assume not-P is true? It's false. We can't assume it as true. It's logically impossible. For example, it's logically impossible for the square root of...