Proof by Contradiction is a method of proving the validity of a proposition by assuming that the opposite proposition is true when doing so would lead to a contradiction.
However, my concern is that the contradiction so reached only proves the invalidity of the opposite proposition; it does not guarantee that the contradictory conclusion reached necessarily determines the truth of the opposite proposition. The best example I can think of is the statement of the falling bodies in a gravitational field (of course, this is not a mathematical proposition, but it does very well highlight the problem). So Aristotle was the first person to claim that heavy objects fall faster than lighter ones, but it took Galileo to theoretically prove that this cannot be true, and his argument runs as follows:
If an heavy object falls faster than a lighter one, then when the lighter object is combined with the heavy object, the lighter object should retard the motion of the heavier object. But the objects combined together have a greater mass than the heavier object alone, and so it follows from the the initial premise that combined objects should fall faster than heavier object, which is a contradiction.
Even though the argument convinces us that Aristotle's claim is false, it does not tell us what the true statement is, and this can easily be understood if we reverse the initial premise. Let's now assume that heavier objects fall slower than lighter objects, and, following the same line of argument as above, it follows that the heavier object when combined with a lighter object should retard the motion of the lighter one, and this would be consistent with their combined state, because the combined objects would then fall slower than themselves individually, which agrees with the premise.
In mathematics, a simple application of the proof-by-contradiction is to prove that the [text]\sqrt2 is irrational, for example. Of course, it would be very bizarre to assume that square root of two has some other property than just being irrational that the proof fails to take account of, but would it not possess a property that somehow would explain its irrationality?
The apparent hopelessness in science is overcome by subjecting scientific theories to experimental observations, whose refutation demands modification of the theory, but how would mathematics or logic overcome this problem of providing a direct proof for mathematical theorems?