One might flip that over and suggest that mathematical laws are derived from observing physical reality. Newton and the apple, and all that.
One might, but it doesn't work the same way. Note that you say 'observing' physical reality, and unless you have a theory of observation then simply observing physical reality to describe it isn't much good. The key distinction is that "observing" is largely a mental process that cannot be studied empirically. So, when you then try to form a theory about reality without being able to take into account a real but non-empirical aspect of reality, then that theory will lack external consistency.
Just saying "the set of all sets, includes itself" is not all that meaningful. Nor is "fundamental inseparability" without stuff to which such a theory might apply.
You'd be surprised what can be logically inferred from simple statements. 1=1 for example has vast implications.
Fundamental inseparability touches upon the issue of the nature of identity. Wouldn't you consider this an important matter given that you're...well...
you?
The notion of what is subservient to which is not all that meaningful.
It's not? There are entire logical fallacies based upon that (fallacies of hasty generalization and slothful induction). Also, set theory...
Math serves to describe the physical world, so to then suggest that the physical world "obeys" in "subservience" to mathematics is going to raise some questions that may be difficult to answer with anything other than "well, we just haven't discovered all the mathematics yet". All that says is that we haven't yet made observations of the physical world to the level where we have the language to describe it.
See what I said above about the distinction between observation (non-empirical) and physical reality (empirical). All you need to do is realize that the scientific method is essentially one of many theories of knowledge that utilizes a certain set of tools and wields certain assumptions. Then, see that there are other (valid) theories of knowledge that utilize other tools that empirical ones do not, and may even hold fewer initial assumptions (thereby adhering more closely to Occam's Razor).
LHC, E8, and all the rest are on a path to developing that language, but are "subservient" to the engineering effort to make the observations. They serve each other.
Carrying off of your last quote snippet, I
think that part of what you're alluding to is certain limitations of mathematics. For example, the 'undecidable' nature of math makes it difficult to discern whether one model is more or less valid than another in describing something (e.g. which is better in describing human interaction: a behavioral model or a cognitive one?). Because mathematics also has limitations, forming a purely mathematical model of reality doesn't do any good either. A good theory of reality not only needs to account for all of reality, both mental and physical, but it also needs to take into account
theory making. In fact, a theory of theories is a requirement given that any description or definition of anything ever is a miniature theory of that thing. Theories aren't just hypotheses that have been rigorously tested and supported; theories can be logical, illogical, right, wrong, small or large in scope, etc.
Interestingly enough, you touched upon what I believe is the essence of the solution. A certain type of language is needed in order to talk about reality, theories, and even language itself...a 'metalanguage' (not my term).