How Do You Prove Religious Arguments?

One thing that is strange is that people can have religious discussions for years on end without agreeing on conclusions. I would like to talk about how "proving" works in the field of mathematics.  Here is a field where we still use the concepts handed down to us from ancient Greece, without change.
One valuable "fact" that is handed down to us from that time is that there is an infinite number of prime numbers.  Here is how the proof goes:

Suppose there is only a finite number of prime numbers.  For simplicity, I will say there are 3: (p1, p2, p3).  Now let me calculate a new number, N, where N = p1*p2*p3 +1.
Now observe that if you divide N by any of your prime numbers (p1, p2 or p3) you will have a remainder of 1.
Therefore your claim that there is a finite number of prime numbers is wrong, so there must be an infinite number.

So that is the proof.
My point in bringing this up is that this is such a powerful concept, and yet it relies only on reasoning.  There was no blinking light at the end that shows the "proof" is correct.  All we have is the fact that for generations, nobody has been able to find a flaw in this reasoning.  Again, no blinking lights, but what makes it "true" is that no refutation has been found.