In a proof by contradiction, how do we know the assumption is the cause of the contradiction? In a proof by contradiction, how do we know the assumption is the cause of the contradiction?  And not just the result of some other property more fundamental to numbers?  
In other words, how can we be sure we arrived at the contradiction because of the proposition $P$ we assumed, and not because of some other proposition $Q$ we are not even aware of? 

The answer that comes in mind is "Well, if there were some proposition $Q$ causing the contradiction, then it should also manifest in the event of $ \neg P$."
Is it this straightforward or there's more to it?  What if the proposition $Q$ can only manifest in the event of $P$, and not at all in the event of $\neg P$? 
Sorry if this is nonsense - just trying to clear out some confusions. 

A proof by contradiction relies on the fact that ($P$ and not-$Q$) is the negation of ($P$ implies $Q$). The law of the excluded middle, along with the law of noncontradiction, gives that exactly one of these two statements is true, so if ($P$ and not-$Q$) leads to a contradiction it must be a false statement, and therefore ($P$ implies $Q$) is true. Note that if you obtain a contradiction that is unrelated to your assumption of ($P$ and not-$Q$) then you have shown that the universe you are working in is inherently contradictory, we usually like to think that that is not the case.
I would like to mention that the idea that the "law of the excluded middle" is controversial is... perhaps overrepresented in some parts of the internet compared to the opinions you would encounter among mathematicians. I would challenge you to try to, on your own, come up with a mathematical statement such that neither the statement nor its negation is true (or where both are true). I won't hold my breath.
