Maxwell's Demon, Physical Information, and Hypercomputation

The second law of thermodynamics is one of the cornerstones of physics. Indeed, even among the most well-tested fundamental scientific principles, it enjoys a somewhat special status, prompting Arthur Eddington to write in his 1929 book The Nature of the Physical World rather famously:

The Law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation—well these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

But what, exactly, is the second law? And what about it justifies Eddington's belief that it holds 'the supreme position among the laws of nature'?
In order to answer these questions, we need to re-examine the concept of entropy. Unfortunately, one often encounters, at least in the popular literature, quite muddled accounts of this elementary (and actually, quite simple) notion. Sometimes, one sees entropy equated with disorder; other times, a more technical route is taken, and entropy is described as a measure of some thermodynamic system's ability to do useful work. It is wholly unclear, at least at first, how one is supposed to relate to the other.
I have tackled this issue in some detail in a previous post; nevertheless, it is an important enough concept to briefly go over again.

The Universal Universe, Part II: ...but is it?

I have ended the previous post with the encouraging observation that if the universe is computable, then it should be in principle possible for human minds to understand it -- the reasoning essentially being that each universal system can emulate any other. But the question now presents itself: is the universe actually computable?
At first sight, there does not seem any necessity for it to be -- after all, computation and computational universality may be nothing but human-derived concepts, without importance for the universe as it 'really is'. However, we know that it must be at least computationally universal, as universal systems can indeed be build (you're sitting in front of one right now) -- the universe can 'emulate' universal systems, and thus, must be universal itself (here, I am using the term universal in the somewhat loose sense of 'able to perform every calculation that can be performed by a universal Turing machine, if given access to unlimited resources'). Thus, the only possibility would be that the universe might be more than universal, i.e. that the notion of computation does not suffice to exhaust its phenomenology.
And indeed, it is probably the more widespread notion at present that the universe contains entities that do not fall within the realm of the computable. The discussion is sometimes framed (a little naively, in my opinion), as the FQXi did recently in its annual essay contest, in the form of the question: "Is reality digital or analog?"