I think Occam is right: there is no spiritual world. If you say there is, even before you prove that it exists, you should explain what you mean by it. Would ‘spiritual entities’ have influence on us? Do they cause effects in our ‘material world’? If they do, why are they not open to scientific investigation? In my opinion there is only a spiritual way to face our life. And being open for scientific facts, and being fascinated by them, is certainly an important part of it.
Further you say:
Theology is like mathematics, not science. Mathematicians start with axioms (initially accepted truths) and use logical derivation to justify consecutive claims, called theorems. Once proven, a theorem cannot be rejected, unless a logical error is found in the derivation.
I think this is completely wrong: you leave out the fact why mathematics is so important for science, and theology is not at all. Mathematical structures are found in all kinds of logical structures in nature. This makes mathematics so useful in science: when I find a group of phenomena that behave like some mathematical structure, I can derive new empirical facts from them according to mathematical rules.
While mathematics does not necessarily has a connection to reality, its usefulness is per definition that many laws of nature can be described with mathematical structures. Theology has nothing similar to offer. It has nothing to do with any reality, except human fantasy.