Irrelevant to science and also outside the capability of science to know. Which was my point in reply to the question. The questioner was asking of the same requirement of a sphere of knowledge outside of science.
Not necessarily - wouldn't you consider maths a science? Theoretical physics?
Higher level math dealing in symbol manipulation - no, this does not completely require reference to reality detectible by the senses. It can be purely datum from the intellect. It is a special and very interesting case that illustrates some fundamental truths about knowledge.
Theoretical physics – somewhat in the same category although it deals with the requirements of realit that I described in my definition, only in a theoretical rather than empirical method. [For example theorizing about what particles must/could exist in order to maintain symmetry through transformations of matter.]
Both these examples tend more to prove the rule by their exceptions and both cases have their applied or experimental application. In the case of theoretical physics this necessary for them to accomplish the scientific requirement of "proof" or "exists." [e.g., Dirac's theoretical work was only "proven" by experimental evidence of the positron.]
I'm not sure what you mean by the "rest of reality".
Those aspects of reality, things that exist, that cannot be known using science alone. We all know a great many of them, some are the most important aspects of being a human.
And I enjoy living in a universe governed by science
This is not a very scientific statement. Science doesn't "govern."
rather than the whims of a temperamental deity.
Perhaps you've chosen to accept someone else's description of diety rather than explore, test and know for yourself. Again, not very scientific.
Math is an art, not a science. Math comes into the realm of science only when it actually measures or discribe something.