First, this whole forum is dedicated to Astronomy and, as I remember the division of fields in most universities, Astronomy is usually a specialty of physics: i.e., back when I was a student an astronomy degree required a rather significant physics background.
Second, Science in the absence of mathematics usually requires an adjective to separate it from what are commonly called the "hard" sciences: i.e., such as the social sciences etc. And even in these "special" sciences, an advanced degree usually requires enough math to understand statistics at least. So, science in the absence of mathematics seems to me to be a concept reserved for grade school level science.
Finally, as far as I am aware, most serious religionists would take it as an insult to suggest that logic is not required in a discussion of god. Thus I find the idea that a "God and Science" forum should be constrained to disallow the use of mathematics or logic is equivalent to reducing it to grade school level of correspondence.
To quote Richard Feynman, mathematics is the distilled essence of logic. Why would you expect understanding things not to require logic. My work uses numerical representations of undefined things mainly so I can deal with massive numbers of undefined concepts to discover a rational self consistent interpretation of all of them. Only mathematics provides an internally consistent structure large enough to bring so many diverse things into alignment with one another.
You speak of a thought crossing your mind. How can a thought cross your mind without being defined? How did you achieve that meaning? How do you know that your interpretations of those words you use are consistent with your experiences of their uses? Does another interpretation of the meanings of things exist which will bring all your experiences in alignment? How do you know such a reinterpretation does not exist?
Words are all defined in terms of other words. Any description of anything is eventually circular if taken far enough! The only thing which might be meaningful is the collection of consistent paths through those definitions. You take a short romp down one of those paths and you get an emotional feeling that you "know what you are talking about". So long as the paths are sufficiently long that the fact that they are eventually circular is not obvious, you can maintain that emotional conviction that you are saying something of value.
Take the big picture for once. For the last 2300 years, great thinkers have been trotting around in that great garden of conceptual paths, constructing mental models of pieces here, there and everywhere. It is their fond hope to eventually learn enough to put all this together and achieve the reward of understanding it all. The problem is, the whole thing is circular; there is no starting point!
So what do I propose to do about it? Well, I will start from exactly where everyone else starts: I will presume I know what I am talking about! However, I will ignore all the poorly modeled paths and rely only on the ones which seem well modeled: "logic" and "mathematics". I will use these well modeled concepts to create a model of my own design.
The only constraint I place on my model is that it must provide an explanation of an enormous volume of totally undefined things (they are undefined if I really don't know what I am talking about). Think of my model as a designed procedure for keeping track of those undefined things (those numerical labels). It is entirely arbitrary just as the Dewey Decimal system is arbitrary. However, if you use my system, certain well defined collections of labels are guaranteed to obey some rather unusual rules. Those rules just happen to be what are ordinarily called the laws of physics. What this means is that any internally self consistent explanation of anything may be mapped into the rules of basic physics.
So, getting back to your explanation of god. I have proved that if you can provide me with a complete coherent internally consistent explanation of god (answering every question I ask on every subject available until you are convinced I understand everything you have said), then there exists an interpretation of the vocabulary you used which is 100% consistent with your every usage of those words and which is also 100% consistent with a foundation identical to classical physics: i.e., in that alternate interpretation, god will be some physical concept obeying all the laws of physics. That result flows directly from the fact that I can take the entire conversation I had with you, apply arbitrary numerical labels to every word used in the exchange and end up generating an explanation which will be totally consistent with what has happened and will yield exactly the same exchanges achieved in the original conversation (including that part of the conversation which convinced you that I understood what you were telling me).
Now, does the explanation of god have anything to do with mathematics or not?
How much math does one need to understand the arbitrary construct I have created? One needs matrix algebra and calculus up through partial differential equations.
What do you need to understand philosophical issues? I don't know as I do not understand the philosophical issues. But I would suggest that the ability to understand all the subjects you mention should be required. In fact, since philosophy seems (at this time) to be somewhat short of serious answers to serious questions, I would suggest that all philosophers should be well versed in all mathematics concepts so that the possible existence of a concept important to their thoughts will not be overlooked.
I have not claimed to have solved all questions. I only claim to have set up a very valuable and dependable foundation entirely through deduction with absolutely no use of induction. The things yet left to be thought about are far more extensive than those conceived of in your view of the universe.