In mathematics, claims must be supported by a process of deductive reasoning known as a proof. "This looks right" does not suffice as a proof; there needs to be a clear sequence, however long or short, of steps that establish the claim as true. Proofs are one of the most difficult but also one of the most important components of mathematics: without them, math is reduced to subjectivity and baseless assumptions, defeating its entire purpose.
One of the more unique types of proof is known as proof by contradiction. It works like this: Take the statement that you are trying to prove, in if-then form (if X happens, then Y has to happen). Assume the logical inverse of the result, our "Y." Through a sequence of steps, show that this assumption eventually leads to an impossible result--the contradiction. That means that the assumption was wrong and "Y" was right all this time.
Follow me below the fold for an example of this (which I will try to explain in layperson's terms, defining any jargon as needed).
Remember the Pythagorean Theorem? a^2 + b^2 = c^2 always works for a right triangle--a triangle with a right angle in it, and long side, c, called the hypotenuse.
A right triangle.
This claim can be proven by
a number of different methods. The processes may differ, but the overall result is the same: a^2 + b^2 = c^2. Give me the lengths of two of the three sides, and I can use the Pythagorean Theorem to find the remaining side.
Inherent in the setup that a, b, and c must all be positive real numbers (the type of number that most people know). Otherwise, we get into imaginary number shenanigans, and an ordinary triangle can't deal with that. Also, the hypotenuse, c, must be the longest side of the triangle, something we can prove by contradiction. Here is how that works: We'll start by assuming a 100% false claim, c <= b (i.e., less than or equal to). Then we'll construct a series of steps and show that that leads to an impossible result. Conclusion? The initial assumption had to be wrong. The following is an illustration of that process. (Note: If you're not familiar with proofs, then take this slowly, waiting to proceed to the next step until you understand what's going on so far.)
1. Assume that c <= b. (We'll consider the case c <= a later.)
2. From (1), c^2 <= b^2. (safe to do because c and b must be positive. Put in sample positive numbers for c and b into (1), and you'll see that (2) always works.)
3. Subtracting, c^2 - b^2 <= 0.
4. a^2 + b^2 = c^2 (the Pythagorean Theorem).
5. Subtracting, a^2 = c^2 - b^2.
Notice that steps (3) and (5) look a lot alike. So let's combine them.
6. Substituting (5) into (3), a^2 <= 0. This includes two possibilities: a^2 < 0, or a^2 = 0.
7. If a^2 < 0, we have a contradiction. Remember the given setup that said that a, b, and c had to be real numbers? It is impossible to square any real number and get a negative result. It cannot be done. (-3)^2 = 9, (-1.5)^2 = 2.25, etc. There is no real value of a that can possibly make a^2 negative.
8. If a^2 = 0, then a = 0. But this is also a contradiction. Again, the given information dictates that we need to have positive-length sides, and zero is not positive.
9. Combining statements (7) and (8), then statement (6), a^2 <= 0, is a contradiction.
10. Therefore, the assumption in statement (1), c <= b, was false, forcing the only other possibility--c > b. Thus it is proven.
Proving that c > a is very similar to the above. Simply switch a and b.
Could I have proven the claim directly--i.e., without contradiction? Sure:
0. Recall, a^2 + b^2 = c^2, and a, b, and c are all positive.
1. From this given info, a^2 = c^2 - b^2.
2. Since a is real, a^2 is positive.
3. From (1) and (2), c^2 - b^2 > 0.
4. From (3), c^2 > b^2.
5. From (4), and since c and b are positive, c > b.
Notice that in this case, it was faster to just do it directly. That happens sometimes. But not always. Without getting into technical examples, sometimes direct proofs are much harder than proofs by contradiction. It may take forever to set up a series of statements that prove the desired claim, but by starting with an intentionally false claim and then knocking it down, we may be able to take a shortcut.
Now how can we extend this idea to debate in general?
One of the curious properties of a proof by contradiction is it can "hook in" people who do not believe the claim however true it may be. So by starting the proof off with the assumption that your claim is wrong (and thereby theirs is right), they may believe you're considering their side. Only you show that the faulty assumption leads to a contradiction.
An example of that might be the blatantly false claim that Saddam was harboring "weapons of mass destruction." So assume that, and show that it leads to an impossible result. How? By considering that even after we invaded Iraq and assumed authority over the entire nation, we were unable to generate any evidence of the sort. So we conclude that the initial claim--Saddam was harboring WMDs--was false.
Of course, the only real drawback is that all parties involve understand and respect the basic tenets of logic. People who choose not to do that are going to be unwilling to have a civil debate because of their close-mindedness. They are Ferrous Cranuses, through and through.
But there is a silver lining: Once somebody wakes up to the truth, it is very hard to lull them back to sleep. Once they realize that Saddam had no WMDs, or that sexual orientation is not a choice, or that the Apollo 11 moon landing actually happened, or that Obama is not a Kenyan Muslim socialist facist, then it is a lot harder to re-deceive them. And we're slowly but surely winning that battle. We may be going through a temporary setback as the deceivers and their followers fight back for the control that they are losing, but deep down, they know they are losing and cannot recover. Indeed, their contradictions are being exposed, and there is no way to stop this.