Reductio ad absurdum

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Hi, I have a little question concerning contradictions :

If I have a statement "A" that I want to prove, and only have the possibility for it to be True or False.

After some manipulations, I arrive at some contradiction. (Here's where my question begins.)

How can we know that a contradiction is enough to be sure at 100 % that a statement is not correct?

Is it because in Mathematics, for a thing to be True or False, it must always be ALWAYS "working" without arriving at some contradiction ? (Mathematical ideas must always work, and not sometimes yes, sometimes no.)

I just want to be sure of thinking of it in the right way, corrections would be greatly appreciated ! Thank you !
 

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D H
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How can you know that a contradiction is enough? After you've checked to make sure you didn't make a mistake.

Mathematics cannot tolerate even one contradiction. With one single contradiction (X and both not X are true), *all* of mathematics falls apart. It's called the principle of explosion. Wikipedia article: http://en.wikipedia.org/wiki/Principle_of_explosion.

That said, there is no contradiction if you've just made a mistake somewhere. You've made a mistake, that's all. Suppose that you find a contradiction as a consequence of some initial assumption and verify that every step after making some initial assumption is solid. That doesn't mean you didn't make a mistake. You did. The mistake was making that initial assumption! The initial assumption has to be incorrect. By assuming something and then showing that this leads to a contradiction, you have but no choice but to reject that initial assumption.
 
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AHhhhh! This is clear now ! It make sense lol

Thank you !
 

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