Typically in physics we derive laws from principles. For example, the law of conservation of momentum is derived from the principle of translation invariance.

Nobody calls the Standard Model a law, for example. The modern view is that the Standard Model is a low-energy effective field theory.

But, whatever supplants the SM, we still expect the principle of translation invariance to hold.

Until, that is, we have evidence for a paradigm shift. If we discover physics that really can't be described, for example, by dynamics happening in a geometric space, then we'll have to give up that principle. Strongly-coupled stringy dynamics seems to have non-geometric phases, for example.

So our statement of laws is more a description of the current best paradigm (say, the operating system), rather than our best model (the program).

You shouldn’t see mathematics as consistent because they are not.

A system can be complete, meaning all true statements can be built based on axioms, but it cannot be sound leading to contradictions.

Alternatively a system may be sound, but not all true statements could be derived.

And finally there are statements impossible to prove because the proof is undecidable.

You’re quite right, which is why I described it as highly rather than completely or perfectly consistent.

It’s an interesting question whether the physical world is perfectly or merely highly consistent. If it’s made of mathematics, it may be that it cannot be perfectly consistent. So if we ever find that it is perfectly consistent, that might be evidence that it isn’t made of mathematics.

Most likely we’ll never be able to tell, but we’ll see. Or at least maybe our descendants will.

> whether the physical world is perfectly or merely highly consistent

What does that even mean? The physical world is not a formal system in any obvious sense.

Nor does "consistent" apply to mathematics, by the way, only to formal systems used/studied by mathematics. You cannot mathematically prove or disprove that mathematics is consistent or define precisely what that would mean because mathematics itself isn't rigorously defined. If you define it as "whatever mathematicians are doing", I guess it's in some sense inconsistent, since mathematicians often disagree.

I wish I understood it better, but you don’t have to go much below classical mechanics before I’m lost.

But from what I’ve read, the deeper you go, the more it’s difficult to find something other than mathematics. When you ask what is a particle you find out it’s probably an excitation of a quantum field. So what is a quantum field if not a mathematical structure? Is there a physical reality to wave function collapse?

Maybe it doesn’t matter. The shut up and calculate crowd doesn’t seem to care.

The deeper we go the more objects we find that we can only describe using mathematics, for sure. That’s not the same as them being mathematics though. We don’t know what the essential nature of these objects is.

It is an interesting speculation, but it’s also possible it’s just confusing the map for the terrain.

If mathematics is the only way we know to describe them, then they might be mathematics. That seems like the simplest possible explanation, so that's probably why I suspect it's probably the correct.

It also seems like a comfortable answer to the question about about what's happening every time I cause wave function collapse and split the universe. I'm just creating a new mathematical structure.

What do you mean something other than mathematics? It's just a language problem nothing else. Just because the English language lacks the descriptive power it does not mean those objects are "mathematical", just as a round ball is not "English" in it's nature just because you can describe it's properties and behavior using that language. A quantum field is just that, a field, it exists just as much as a magnetic field exists and that you can observe directly. It is an area of space where a specific "force" you might say has impact on objects that enter that area of space. It's properties are measurable and they produce consequences in the world and it has a precise description using the language of mathematics

> It is an area of space where a specific "force" you might say has impact on objects that enter that area of space.

Aren't you describing a classical field?

As I understand it, a quantum field is essentially varying probabilities (the wavefunction).

> they produce consequences in the world

They don't produce consequences in the world, they are the world. You and I, we're excitations of a quantum field. We're of the wave function. The quantum field encodes all the information that is us.