Mathematicians have been at it for thousands of years too, and they've produced a new body of knowledge...
And yet noone has ever been able to provide any kind of proof for say, the 4(+1) axioms that Euclidean geometry is based on(although attempts have been made, similar to, for example, squaring the circle or constructing a perpetual motion machine). Still, no deductible proof. They are taken as true and the whole 'Euclidean Geometry' construct is built on them as such.
So what do you say, Euclidean geometry is illogical, seeing as is it based, as a whole, on statements that cannot be proven? Is it circular? Or just axiomatic? As in, choose those axioms, you get Euclidean, choose others, you get hyperbolic geometry?
Can you prove the parallel postulate? Or any other of the axioms? What's the difference between an Axiom and a theorem? Notice this?
That's silly. Mathematics makes no claims at all about the real world. Mathematics is all about "what-if" type question in entirely abstract logical settings. The fact that conclusions in mathematics are often useful in the real world is just a nice side effect.
The parallel postulate is actually a very good example of that "what-if" thinking. People were uncomfortable with the parallel postulate for a very long time, because they thought they should be able to prove it from the other axioms. Eventually somebody said "Hey, let's try and see what happens if we replace the parallel postulate with something else!" That's how non-Euclidean geometry was invented, and it turned out to be quite interesting.
Let's continue with that theme. Your question "Can you prove the parallel postulate?" is, by itself, meaningless. You have to clarify what you really mean by it. Did you mean to ask "Can you prove the parallel postulate from the other axioms of Euclidean geometry?" The answer to that question seems to be no, given that we have models for non-Euclidean geometries, i.e. geometries where the parallel postulate is false. Another question you could ask is "Is the parallel postulate true in the physical world we live in?", or some variation thereof. However, this is no longer a mathematical question, but a question of physics. It has nothing to do with axioms and theorems, just like the question "Is C12 radioactive?" has nothing to do with axioms and theorems.
There are actually much, much more examples of those "what-if" type questions in mathematics or logic. You may want to read up on the different axiomatizations of sets, for example.
In that sense I would like to disagree with SamLowry slightly. Mathematics is
about absolute truths. It's just that the absolute truths in mathematics are ultimately of the form "If ... then ...", when you really drill down and formalize them, e.g. "If we work in that system of axioms, then such and such is true". Even more important, mathematics is not about the real, physical world we live in, whereas any claims of the existence of a God ultimately are about the world we live in, and therefore different rules apply to them.