> I do not think it a coincidence that our formulations of physical laws are so
> naturally expressed mathematically, I think it means we need to consider the
> possibility that the laws of physics themselves are an extension of
> mathematics.
I'm not sure I agree with that. As a colleague of mine noted recently, while physical laws are continually being adjusted, tweaked, expanded, or sometimes even overturned, properly proven mathematical laws (theorems) have never had this occur (certainly, some conjectures have been disproven, but no properly accepted and rigorously proven theorem has as far as my mathematician colleague is aware). He added that, to him, pure mathematics was as close to the Platonic ideal as one could get.
But let me give you an actual example as well that came up during a conference that just ended. It's a rough description, but it will suffice for what I'm trying to say.
An open question in quantum information theory relates to something known as quantum channels. Pretty generally, these are communication channels and could be just about anything. But some of them have non-unitary behavior meaning we can't approximate them using a bunch of unitary operators. It has been conjectured that if we take more and more copies of one of these channels, we might be able to get closer to an actual unitary representation. Specifically, in the asymptotic limit, the conjecture assumes it is possible.
Now let's put on our experiment's hat. As useful as this conjecture sounds, it means to get a perfect unitary representation we'd need an infinite number of copies of this channel. But this makes no physical sense. No experimenter can ever physically achieve this. Nevertheless, the conjecture exists and may end up being proven.
Another way to put it is that we have plenty of mathematical statements and proofs for situations that are completely unphysical.
Now consider that many - if not most - physical laws can be modeled in multiple ways mathematically, i.e. it is possible to model some physical laws using two seemingly unrelated mathematical procedures. Which procedure would the physical law be extending? In non-relativistic classical theories, there are two types of physical laws: laws of coexistence and laws of succession. In quantum mechanics these become selection and superselection rules. But this puts a serious limitation on mathematics if it is an extension of it since, for example, the former would imply that non-relativistic classical theories are only describable by equations and inequalities and yet there are so many other mathematical structures that can be used to describe even the simplest things in nature, not to mention the fact that mathematics itself has further generalizations for equations and inequalities that work in classical situations (groups, categories, etc.).
I hope that made sense. It's past my bed time.