You have focused on how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics.
You have also taken your thought in line with Hamming.
You are trying to explain which one is more fundamental mathematics or physics?
Its indeed the clue what governs the mathematical equations,operators e.g. addition,multiplication etc. Its certain laws of invariance,symmetry order.Its not mathematics describing physics but the laws of invariance,symmetry,order behind those mathematical structures explaining that of physical reality. If we look at the algorithm of formation of numbers and operators e.g. ,-,*,/, we will find that the particular invariance, symmetry,order is drawn from the physical world reality itself e.g quantum structure of energy levels of atoms.
It is possible that the invariance,symmetry,order of classical mathematical structures e.g.hypotheses of geometry don't resemble the those characteristics of quantum world.This raises the deeper question in context of Riemann's geometry,that hypotheses of geometry don't at conform at quantum level.
So, this coherency and compatibility is the key why mathematics has been so effective in Physics. because my Mathematical Structure Hypothesis states that mathematical abstraction and physical reality both originate from Vibration. This is why sometimes mathematics explains physics and other times physical theories solve the mathematical problems.its a two-way interdependence but the different manifestations of the same Vibration.The mechanism of perception of Integers,addition all are its products deeply to be discovered ,how? Thats why e.g.entropy which was considered to be physical phenomenon also governs the mathematical structures as in Poincare,Geometrization conjectures.
Thats why the clue is to match the laws of invariance,symmetry,order of the two,otherwise it leads to mutual friction. We need to match the physical characteristics of mathematical abstractness with that of physical reality. Riemann Hypothesis is all about this laws of invariance behind functioning of these operators, complex analytical continuation.
In context of Skolem paradox,a particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong.
This is because that when mutual laws of invariance matches each other,it gives right result and otherwise wrong because of mutual friction.
Its equally important that as Wigner said that physical laws are conditional statements. What mathematics tries to do is superficially tries to find out the quantity relations between observables rather than why something happens.
What should be field of further research is as Richard Feyman said- The next great era of human awakening would come ,today we dont see the content of equations.
So,the nest era of research is to discover those laws of invariance which governs the mathematical equations itself and then match it with physics.
Anyway, you have written a great essay .