Ok, some more in depth commentary is required on the matter of mathematics.
What matters in the realm of pure mathematics, is whether or not any equations are consistent with the axioms and prior theorems that led to the construction of said equations. Whether or not those equations happen to model the behaviour of a real world physical system is irrelevant to pure mathematician - what a pure mathematician is interested in, is investigating the behaviour of equations as pure, abstract entities.
In the case of applied mathematics, this is the discipline within which additional criteria are brought into play. The most important of said criteria being whether or not an equation models the behaviour of a physical system of interest precisely and accurately.
Pure mathematics is, of course, the canonical example of a system of formal reasoning, and its practitioners have produced a wonderful, at times brilliantly elegant, and at others shockingly counter-intuitive, collection of theorems within that discipline. But only a few, shall we say, interesting individuals, have ever claimed that the products of pure mathematics dictate how reality behaves. Most recognise that a subset of the products of pure mathematics happen to be uniquely placed to describe how reality behaves, but there is a far larger subset of the products of pure mathematics that don’t describe how reality behaves.
For those requiring an insight into why I wrote that last sentence, I have in my collection, a nice little textbook on theoretical mechanics. The work in question being Theory and Problems of Theoretical Mechanics by Murray R. Spiegel, Schaum’s Outline Series in Science, McGraw-Hill Book Company, ISBN 07 084367 0. Turning to Chapter 5, Central Forces and Planetary Motion, beginning on page 116, we move to solved problem 5.15 on page 127, which requires the determination of the nature of a central force that generates a particular path of motion in space. The path in question is a somewhat peculiar one, and requires the orbiting particle to pass through the point from which the central force generating the orbit originates.
As an exercise in mastering the requisite mathematical tools, it’s pedagogically useful, and the answer to the problem, is that the central force in question obeys an inverse fifth power law - namely, if r is the distance from the orbiting object to the origin of the central force, then that force is given by F = k/f5, where k is some suitable constant of proportionality.
The point I’m making is that this is an example of the many products of mathematical reasoning that don’t describe any observable system. Useful as a theoretical exercise it may be for the student, I am not aware of any observable system that this theoretical construct describes. Indeed, as a proposed model for gravity, is it demonstrably wrong, because, for example, the planets in the Solar System manifestly do not move in this manner. Indeed, if they ever had done, they would only have done this once, for reasons that should be obvious to anyone who has paid attention in a science class, and we would not be here discussing this topic as a corollary of said orbital behaviour. In short, planets obeying this central force law would crash into the parent star on the first pass, for those who haven’t studied the requisite topics.
Quite simply, what holds true in an abstract world described by a formal axiomatic system, need not hold true in the concrete world. A fortuitously useful construct such as a k/r2 law, that happens to be an excellent description of the behaviour of gravity (and for that matter, electromagnetism), may arise from a formal axiomatic background, but we have no guarantees that said formal axiomatic background will always deliver the goods. Unfortunately, the caveats applicable here never register with the usual suspects.
Physicists (and other scientists, for that matter) have known for some time, that it doesn’t matter how beautiful, elegant or sophisticated your theory is, it can always be brought crashing and burning by a set of data that says “no, you’re wrong”. Indeed, a parallel caveat applies to formal axiomatic systems even when we don’t try applying them to the concrete world - namely, that it doesn’t matter how beautiful, elegant or sophisticated your formal axiomatic system is, it can always be brought to its knees by internal inconsistencies, or, in some cases, the Incompleteness Theorem.
At this point, I shouldn’t really need to tell astute readers that the incompleteness Theorem constitutes, in effect, a “barrier of ignorance” that no axiomatic formal system can ever cross, and, furthermore, the more powerful, sophisticated and expressive your formal system is, the more it will fall victim thereto. But it’s still worth mentioning, because the usual suspects are either unaware of this, in which case their attempt to conjure entities into existence is incompetent, or pretend that their pet system of “reasoning” is exempt therefrom on grounds that are usually revealed to be specious, in which case the practitioners of this brand of legerdemain are duplicitous and not to be trusted.
That paragraph above doesn’t negate the results shown to be consistent within the formal system in question (another mistake the usual suspects make, frequently doing so deliberately in order to facilitate mendacious apologetic fabrications). What the Incompleteness Theorem actually states, is that there exists at least one proposition that is true within the system, but not derivable within the system. That proposition is still true - it’s just unreachable unless you move to a different formal system where the proposition in question is derivable. In layman’s terms, a Gödelian-undecidable proposition is one that is true within the system, but which your system can never alight upon.
There are, of course, subtle technical issues applicable to formal systems, with respect to whether or not they are axiomatically “strong” enough for the Incompleteness Theorem to apply, an interesting outlier being real analysis, which escapes the provisions of the Incompleteness Theorem in part because it admits of quantifier elimination. But expuinding upon that would require spending two decades delving into the minutiae of whether or not the Löwenheim-Skolem Theorem applies to your formal system, a task beset with intimidating hurdles even for skilled mathematicians. Needless to say, I don’t propose even to try taking a detour down that alley - a successful exposition would make me a Fields Medal candidate for one thing …
In short, for every product of pure mathematics that happens to possess utility value in modelling observable physical systems, there are thousands, indeed possibly an infinite number of possible products thereof that don’t exhibit this utility value.