1 - P=NP is a mathematical problem. It has nothing to do with Physics. Physics has to do with Mathematics but one should be very careful when extrapolating (range, constraints, etc).
2 - Nature has no problem whatsoever solving complicated equations. Our mathematical models are the ones who suffer to model simple everyday stuff in Physics. Turbulence and Navier-Stokes equations, electromagnetic propagation, the way lightning goes through the air, etc.
Unlikely, but cute...
The granularity of the simulation of our reality is the Planck length. This constraint does not apply to the real, underlying, reality within which our simulation is embedded.
I don't know if I accept that, although it's unclear what exact definitions you're using for those terms. P=NP makes very real claims about the abilities of real physical objects like Turing machines. It's obviously about math as well, but I don't see how that precludes it from being about physical qualities of physical systems.
> 2 - Nature has no problem whatsoever solving complicated equations.
Solving some complicated equations, no doubt. But I'm not aware of any evidence suggesting that nature can easily solve all complicated equations.
Or at least any physical approximation of theoretical constructs like Turing machines.
There are some who would disagree with you. It is quite arrogant to assert that computer science has little to do with physical reality when our constraints on computational capabilities are very much embedded in reality.
Take a look at this survey article, for instance: http://www.scottaaronson.com/papers/npcomplete.pdf
There is no evidence that nature has NP-hard problems. All the nature is built on local interactions - thus the finite speed of any interactions in the nature.
Note: the only example of non-local optimization (specifically maximizing entropy over time-space existence path of a living object vs. that would be an entropy change associated with the equivalent amount of non-living matter in similar environment) is the life itself.
P=NP is not just statement about difficult problems or big equations, it's a very particular class of problems.
Regarding "Hypercomputation" in particular, I've typically encountered it in the context of "solving problems a TM can't solve" rather than "solving problems a TM can solve but asymptotically faster" - is it actually used for both? A skim of the article didn't clarify.
Of course, an Zeno machine, or a machine that can solve the halting problem, could also certainly solve problems a TM can solve but asymptotically faster.
I totally agree, which is part of why the article didn't clear it up.
"Of course, an Zeno machine, or a machine that can solve the halting problem, could also certainly solve problems a TM can solve but asymptotically faster."
A Zeno machine, for sure. It is not immediately clear to me that this necessarily holds for anything that can solve the halting problem - what about a TM plus a halting problem oracle that would tell you in 2^(size of TM) whether a TM halts?
Of course the real question is whether asymptotically faster is enough to be "hypercomputation". Necessity is also interesting, but not the same thing.
To say that a mathematical problem has nothing to do with physics would not be right, since physics' major theories of today are essentially maths.
No
Physics depends on mathematics, not the opposite.
Math exists regardless of physics.
Yes. Our theoretical models of the laws of physics are usually formal and axiomatic systems of logic (basically mathematics).
Yes. We understand and describe it using mathematical constructions we know of.
However, the laws of Nature do govern what kinds of models of computation are realizable (theoretically and physically). Limits on computability influence the design and sophistication of logic and mathematics. A good model of computation that we choose for this is some variant of a Turing machine. If this is true for our brains as well, then there is a limit on how sophisticated and powerful mathematics can be from our perspectives. In other words, the laws of Nature are dictating how good of a system of mathematics we come up with can be from a logical standpoint.