That's such a cool question because sometimes the answer is no today and yes tomorrow.
The most striking example is number theory, which for over two thousand years has been regarded as a supremely beautiful and supremely useless branch of math. It was the great English mathematician G.H. Hardy, played by Jeremy Irons in the film The Man Who Knew Infinity (a must see), who said that a branch of math was beautiful exactly insofar as it was useless!
He said -- and this was a brag and not at all humble! -- I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.
Hardy's specialty was number theory. Wouldn't he be surprised to know that his belovedly useless number theory is today the foundation of all Internet security, e-commerce, and cryptocurrencies. Number theory is the basis of pubic key cryptography, the idea that makes Internet security possible.
Number theory had no connection whatever with the physical world for two millennia; and became one of our most important practical technologies just in the past thirty years.
So the answer to your question is ... we never know. As another example take non-Euclidean geometry. For thousands of years it was called impossible. In the 1840's it was found to be mathematically consistent, but nothing more than a curiosity. And when Einstein came along, it turned out to be the mathematics of our universe.
Maschke has further clarified my post by practical examples, which I hope to extend with some thoughts. It seems that the interchange between math and physics alternates within and without all limits of both.
An applicable question might be: is there a state in math which corresponds to each event, hypothetical or not, of the physical universe, and vice versa? Is information represented as completely by numerical as mechanical statistics?
However, physics may require an observer to interact with where mathematics may not. Does one define an observer equivalently in both the intangible and material world?
One thing to remember is that any physical system ultimately has an energy associated with it and that bounds things pretty seriously with respect to what mathematics might produce.
Any physical system in our universe that exceeds a certain amount of energy density simply collapses into a black hole into which no observations can be made.
So I suspect the bulk of dynamic systems that mathematics can describe simply disappear into black holes rather quickly.
What more (if anything) is lost to a black hole singularity -- mathematical or physical information? The no-hair theorem says that every physical and mathematical observable variable besides angular momentum, mass and charge is "lost" there.
Well.. I think you're asking one of the big unknowns in current day physics. As I understand it unitarity in QM insists no information is ever lost.
Yet any information associated with quantum objects beyond the event horizon is lost, or at least hopelessly scrambled when it becomes observable as Hawking radiation.
I don't think this issue has been definitively resolved yet.
This is an absolutely brilliant question, and Maschkehas brought to light very clearly the dynamics of it.
If the OP was simply asking if there is something that is out there that wasn't yet described by mathematics so that they could be the first to do so, then we rearrange the question to find out another way.
Is there a possible physical system that mathematics has proven inapplicable?
That could better help in finding an answer to the search.
Now if the OP was using a more philosophical approach, then the question implies the existence of a relationship. I'm single BTW
Existence proof. Many interesting and important theorems have the form ∃xP(x), that is, that there exists an object x satisfying some formula P.
Suppose U is a universe which is appropriate for Mathematics. To prove the statement, there is a function f such that f′=f,
f(x)=e^x works (as does any constant multiple of e^x).
Hilbert's Nullstellensatz, from a philosophical point of view, is interesting, as it implies the existence of a well specified object.
If are polynomials in n indeterminates with complex coefficients, which have no common complex zeros, then there are polynomial such that
When it was introduced, non-constructive existence theorem was such a surprise for mathematicians that Paul Gordan wrote: "this is not mathematics, it is theology"
I would say that no physical system can exist that could not be explained by way or some form of mathematics.
Whether such math or physical system exists today involves the variable of time.
"sometimes the answer is... no today and yes tomorrow"