No, we do physics this way because we don't know any other way. When this way fails (e.g. quantum gravity, turbulence), we assume that the problem is that we haven't tried hard enough yet. Which may be true, but it's also possible that the computational approach simply doesn't work for those phenomena.

This sounds an awful lot like "God of the gaps" argumentation with "non-Turing hypercomputation" substituting for "God". If the computational approach doesn't work — that's all of human reason, to emphasize: any symbols you can scribble on paper and reason about over coffee, that's a finite algorithmic process — then there's no humanly-achievable endeavor that can solve the problem. It's forever hiding in the gaps, the shadows; it's definitionally unknowable.

So what field of science are we discussing right now? One of the more frivolous branches of philosophy: talking about something that we can't define, can't reason about, won't ever be able to reason about; something which marks no experimental evidence in our human sciences. Therefore: it doesn't exist. Apply your favorite razor. Something we can't possibly talk about, let's not talk about. Something whose existence or non-existence are indistinguishable, which leaves no shadow in the observable universe: it does not exist.

It's not a "God of the gaps" argument because I'm not claiming that "God" (that is, incomputable physics) exists. I'm simply pointing out that the fact that our entire understanding of physics is built on a computational framework doesn't actually say anything about the Universe, merely about the way we do physics. There is a priori no reason to expect that the Universe would be in any way amenable to human understanding. We have just found that in some cases, it is.

And non-computability doesn't make something philosophical or mystical. We have encountered phenomena beyond the reach of any computation in mathematics and computer science many times. They exist, no matter which "razor" you apply.

- "We have encountered phenomena beyond the reach of any computation in mathematics and computer science many times. They exist, no matter which "razor" you apply."

The razor applies as to whether the physical universe runs on laws that don't fit in Turing machines. It applies because it's an unfalsifiable proposition. You can't embed a Turing machine in a super-Turing universe, ask it "what is the nature of your cosmos?", and have it determine (within finite computation) there are more things in heaven and earth than it can compute.

(This is the ordinary Church-Turing thesis: if some finite interaction between Turing observer and super-Turing universe convinces the Turing observer of the super-Turingness of its universe, within finite time... then an ordinary Turing machine can also do exactly the same thing, by simulating the first Turing machine, and brute-force enumerating every possible interaction within that finite bound. The nature of super-Turing machines is incomprehensible to Turing machines; there's nothing a super-Turing machine can do to prove it exists to a mere Turing machine).

You're conflating computability with computational complexity; that seems to the source of most of your confusion.

I don't remember the name of the complexity-theory analog of the Church-Turing thesis; but it's an open problem what types of computers physics can support the existence of, how they relate to classical Turing machines in complexity theory. See for example: quantum complexity theory.

About halting: there is provably no way to verify an oracle for the Halting Problem, even if you had one. You can't even ask a hypercomputer to write a proof for you: in general there exist no finite-length proofs.

As someone who works in a custom fabrication shop and who has been research job shop scheduling for over a decade as an attempt to get better at it, it's a computationally impossible task. There are huge numbers of "optimal" ways to schedule the shop and all of them require enormous amounts of computation. Genetic algorithms are offering the most promise right now, but the schedule has to be constantly updated because things never take as long as you think they will.

This is primarily because, unlike manufacturing, the basic unit of our system is a person, with all of their chaotic outputs. You start stacking those up and you end up with "long, fat tails" in every analysis you do. Your probabilities are spread almost flat over a broad range of outcomes.

It's fascinating and definitely falls into the category of uncomputable.

I'm not clear what you mean about turbulence? Quantum gravity involves as-yet unknown physics but turbulence is surely just a case of "the number of calculations we need to simulate it grows quicker than we can reasonably keep up with"? i.e. simple toy simulations work fine but we don't have a planet-sized computer to do something more extensive.

(Just to clarify - this isn't a "classical vs quantum" thing - it's a "we know the equations vs we don't" thing. I'm sure QED simulations are fairly similar within a known domain)

Presumably the GP refers to turbulence being infamously difficult to model analytically. But we have a great model for turbulent flow, the Navier–Stokes differential equations, it’s just that we can only solve them numerically in all but the simplest of cases. But when we do we get good results, so it’s not like turbulence is some mystical phenomenon beyond the reach of our standard mathematical tools!

My knowledge is a bit weak but aren't there non-analytic ways to model turbulence? "Analytical" isn't the whole of maths.

(I might not have a clear understanding of what "analytic" means so please correct me if I'm talking hogwash)

The alternative is numerical approximation, as I mentioned. Like always in physics, any analytic (ie. "closed form") solutions would also necessarily have to be special-cases, approximations, and simplifications. The problem with turbulence is that it's so chaotic and complex that it doesn't seem to be reducible to simpler models while still retaining some predictive power. But that's not very surprising; we're talking about the chaotic motion of molecules at the Avogadro scale! After all, we can't even write closed-form solutions (or even approximations) to the motion of merely three bodies under gravity, never mind ten to the power of twenty-three.

But what is fascinating about turbulence, though, is the "edge of chaos" – the boundary conditions at which laminar (non-turbulent) flow suddenly turns turbulent. On one side of the boundary analytic treatment is possible, on the other it is not.