I studied mathematics in university because I thought that it would be like peering into the mind of God. There is a certain level of rigor required to study it effectively, and I was not particularly rigorous. Any serious math education involves proofs and logic more than memorizing and applying formulas. This textbook, "Baby Rudin" is practically required reading, though reading it is hardly sufficient, one must suffer through the various exercises left to the reader. I was more interested when there were applications in physics and computer science. There is certainly a degree of giftedness involved, but that is mostly relevant at the boundaries of human comprehension. I don't think that those without mathematical knowledge are missing out on any spiritual insight, despite claims to the contrary.
It is difficult to avoid the foundation of much of contemporary mathematics, Zermelo-Fraenkel set theory with the Axiom of Choice, and its inconsistences shown by Gödel's second incompleteness theorem. ZFC is the most common foundational set of axioms, but Gödel has shown that a consistent set of axioms for all mathematics is impossible. Does that mean that everything we know about mathematics is wrong? Not quite, since theorems based on ZFC or any set of axioms are at least consistent within the established system. It does raise interesting questions on what is knowable via mathematics alone.
When mathematicians make claims to elegance and beauty, they are not referring to the supple bosoms of beautiful women. Mathematical elegance is emergent, it is when things which seem impossibly coincidental line up perfectly. Take for example, Euler's formula: eiπ + 1 = 0. It establishes a relationship between zero and one, the natural logarithm (e), the imaginary number (i), and the circumference of a circle (π). There is also an utmost simplicity to it, making it accessible and comprehensible to an educated lay person. This is contrast to things such as the Poincaré conjecture which may seem simple although the definitions used require deep understanding, and the proof of which is not simple at all. When it comes to proving the existence of God, I don't think that one could ascribe a mathematically beautiful quality to it, as I think it would be fundamentally unknowable.
Srinvasa Ramanujan
To me, perhaps the most fascinating figure in mathematics is Srinvasa Ramanujan. He exemplified the amateur mathematician and auto-didact, something which is frowned upon by academia today. He is cited for saying, "An equation for me has no meaning unless it expresses a thought of God." I think that God Himself revealed secrets to him, because how does a mere mortal come up with things like this:
In his time, computers as we know them did not exist, there were no means to brute force something like this. He recorded dreams in which knowledge was presented to him, which he then committed to writing. Other than being born into a Brahmin family, there was not much else remarkable about his upbringing, he had no mentor until much later in life, when he was discovered by the English mathematician G.H. Hardy.
Ramanujan claimed that all religions seemed equally truthful to him, and didn't publish any proofs of the existence of God. Perhaps the existence of God was self-evident to him, and required no proof. Others remain unconvinced.
Ontological Proofs of God's Existence
There is a long and storied history of attempted proofs of the existence of God, perhaps starting with St. Anselm of Canterbury who first proposed the ontological argument in 1,078 A.D. I am squarely in the camp of people who believe that the existence of God is unknowable, and pontificating this question too much leads to no answers. This makes me an agnostic.
Mainly, I don't think that formal logic can express the existence of God Himself. It is a question that is left up to faith alone. Ontological arguments can prove the existence of something, but I don't think that something is a deity. Moreover, I see them as being mainly tautological, because if God represents all that is good and true, how could that not be a tautology?
Quod erat demonstratum which means that which was to be demonstrated, often appearing at the end of proofs, wouldn't really apply to an ontological argument as would be impossible to demonstrate the existence of God without an act of God. What even constitutes an act of God, is up for debate, but I would assume that it would be something like a statistical improbability, like Maxwell's demon decreasing entropy in a system.