The OP asks whether "mathematical structures" like equations exist outside the brain. I say no. Mathematics and logic are systems we invented to help us (i.e., the brain) reason about properties and quantities we can observe. The fact that those things we observe and "measure" (itself a nontrivial concept) exist on their own does not mean our frameworks of reasoning about them exist on their own. Those systems let us deduce what we would observe and measure. That's eminently a brain activity. That's still a big topic, but probably wants to wait for its own post.
It's important to remember that philosophers define "natural law" differently than physicists.
We deduced that the thing we call matter and the thing we call energy are actually two forms of—something. We made some machines that let observe them change from one form to another.
We observe that an attractive force exists between two chunks of the stuff we call matter: the bigger the chunks and the closer they are together, the more force.
We observe that some funky bits of matter exhibit attractive and repulsive forces, and that breaking one of them in half gives us two examples of the funk, not half the funk in each half.
In our special language, we down these observations respectively as E = mc2, F ∝ m1m2/r2[, and ∇⋅B = 0.
In physics, natural law simply describes what can be observed. We initially observe that two chunks of matter move toward each other as if some attractive force were present. As we observe more carefully, we can describe our measurement of the quantities involved in terms of something we call multiplication. It accurately describes what happens but it doesn't try to guess why it happens.
As expected, philosophy wonders if that "why" means something more than physicists care about. The OP asks whether natural law "governs" the behavior of the universe. A physicist interprets "govern" here to mean "describe." We don't make natural law; we discover it. Once discovered, that law describes what we think about how nature will behave. It lets us accurately describe things that haven't yet happened without worrying about why they're going to happen that way.
Some philosophers interpret "govern" here to mean "causes." Two chunks of matter wander toward each other because some separate, higher manifestation of reality made them do it. It implies a causation that isn't formulated as just some inherent property of matter. This poses a problem for philosophical "physicalists." Physicalists say every behavior in the universe depends on matter. This is the basis of saying that the hard problem of consciousness will eventually be solved by looking harder at how matter (especially brain matter) works.
But if there are rules that create behavior in matter, then those rules must therefore exist (in some way) separately from matter. If not everything depends on matter, then physicalism is wrong. The "govern means causation" camp defines natural law as some active influence that makes matter do what it we see it do.
Naturally, as in all philosophy, there are problems with this view.
There's a self-existing law that prevents cars from going faster than 25 mph down my street. Cars do it all the time, meaning that the law has no actual power to compel cars to go that speed. The law is not an elementary or emergent property of a car. The law exists by itself as a separate thing whether there are cars on my street or not. It has an effect on cars only insofar as intelligent drivers of those cars fear being punished for disobedience.
So what property of matter compels it to obey these supposedly self-existing laws instead of just doing whatever it wants? If natural law exists separately from matter, by what mechanism does it affect matter and force matter to do its bidding? What property of matter allows it to be manipulated in this way? Where and how do these laws exist?
So much for mathematics and natural law. What about information and quantum mechanics?
Philosophers are stuck here because if you want to hide dualism inside of quantum mechanics, you need to use the definition of "information" as it occurs in quantum mechanics. The basic element of information in QM is inherently a property of matter. The spin of an electron is information. It's largely navel-gazing twaddle to talk about spin as something separate from an electron. The abstract notion of electron spin can be discussed as a res mentis, but there's no actual information unless there's an actual electron to spin "up" or "down." The polarization of a photon is information. But while the abstract notion of photon polarization exists separately from an individual photon as a "thing of the mind," the information is present only when a photon is present. Information in QM is the absolute manifestation of physicalism.
