IANAM - but - flipping through a book of lay mathematics this example (used in a totally different and over my head context) struck me.
To save me typing let's call the number 10^1000 erth. let's use e in its normal Napierian sense.
Take the statement "The erth digit of e is 4'" This is just engineering or computer science.
Take the statement " In the expansion of e there exists a string of erth 4's." This is true or false and one leg can be shown to be true, although we don't know how long it will take. The false leg cannot be proven by grunt work, we would need some knowledge to say that the statement is false. I think this is mathematics.
Take the statement " the number of 4's in the expansion of e is finite." I don't think there is any way of proving this either true or false.
I think this last statement is religion. To state that either leg is true is a matter of belief, and faith is choosing one option in the absence of evidence, which option for the believer comes to have the force of fact.
So I end as an agnostic, which I have been from the age of 15, since I just cannot jump one way or the other.
When asked I usually tell people that I don't believe in belief, because if an idea is so bizarre that only one person in a million could fall for it, 6,000 people hold that as a tenet of their faith. That said, I do believe in Sidney, AU in Texas, in condors and in the New South Wales Social Sippers.