I think the following paper might be relevant for the discussion.

Artif Life. 2006 Winter; 12(1):117-34.

A category theoretical argument against the possibility of artificial life: Robert Rosen's central proof revisited.

Chu D, Ho WK.

http://www.ncbi.nlm.nih.gov/pubmed/16393453

Here is their abstract:

One of Robert Rosen's main contributions to the scientific community is summarized in his book Life itself. There Rosen presents a theoretical framework to define living systems; given this definition, he goes on to show that living systems are not realizable in computational universes. Despite being well known and often cited, Rosen's central proof has so far not been evaluated by the scientific community. In this article we review the essence of Rosen's ideas leading up to his rejection of the possibility of real artificial life in silico. We also evaluate his arguments and point out that some of Rosen's central notions are ill defined. The conclusion of this article is that Rosen's central proof is wrong.

First, look particularly to A. H. Louie. Rosen's arguments, right or wrong, are profound, relate to complex and nuanced issues, and most importantly are never expressed in some singular, explicit form the way a proof that "(M,R)-systems are necessarily non-computable" require. This is not to demean or in any way detract from Rosen, simply to recognize (as his foremost proponents do) that the development of his ideas over time were never expressed in a formal proof. This is not true of those who have built on his work, hence Louie.

Second, Rosen doesn't argue that physical reality is "non-mathematisability". He argues that living systems are closed to efficient causation, and therefore must have non-computable models. This differs from your description in two extremely important ways. One is the restriction to living systems (Rosen's argument rests on "(M,R)-systems", or anticipatory systems defined in this case specifically in terms of metabolism-repair). This in no ways entails, implies, or otherwise suggests that physical reality itself is non-computable sui generis. It entails that a tiny subset of physical systems are. Another issue is that "non-mathematisability" is not equivalent to "non-computable" or incomputable. We define computability mathematically, which means we can determine what is incomputable mathematically, and this alone entails that what is incomputable/non-computable can be represented mathematically.

That even a portion of physical reality be agreed non-computable is enough to prove the point; t'is the typical mechanism of proof denial in mathematics, is it not? (I need only one case, not sui generis). I understand the distinction between computability and mathematisability; ultimately my concern is with the distinction Rosen draws between mechanisms and life. ATP synthase looks every bit to be a mechanism. If Rosen's argument holds over biological organisms, then non-machines may be constructed from machines. Or, is there in fact a difference between some parts of biological organisms and other parts; that only some parts are alive while the others are just machines? Certainly, ATP synthase is one element of an (M,R) system.

Some points, mostly derived from reading Louie:

@Scalas, apparently the work of Chu and Ho amounts to a straw man fallacy. They misconstrue his definition of 'mechanism.' This is the main thrust of the article: Louie, AH, "A Living System Must Have Noncomputable Models." Artificial Life 13: 293–297 (2007). (I emphasize that I don't really understand the math.)

@Buckley (and also @Messing): the detailed definition of computability is highly relevant. In particular, my understanding is that Rosen equates computability with "computable in finite time" or "decidable." Why should this be contentious? Life processes (and maybe whole organisms) can be simulated, but there's no condition inherent to the simulation that dictates if/when the simulation concludes. Life goes on!

Regarding the distinctions between mechanisms and organisms, I do think the relational biology school asserts that non-machines can be constructed from mechanisms. The mechanisms *collectively* need to satisfy conditions that allow for closure to efficient causation. So reductionists ought not to be threatened from this material -- we can understand ATP synthase itself as a machine. I don't think putting enzymes in your laundry detergent makes your washing machine into an organism!

So, following and analogising from von Neumann, there is a certain critical point of complexity, above which machines become non-machines.

Stuart Kauffman developed that line of thought explicitly. His conclusion was that a randomly wired metabolic network (satisfying the property of being collectively autocatalytic) needed to be very very large before it had a significant probability of accomplishing closure to efficient causation. I think the reference is: The Origins of Order. Oxford University Press, New York (1993) p. 299 (this reference is lifted from the paper cited in the next paragraph.)

While I'm name-dropping (Kauffman's consideration of Rosen's work is one of the strongest endorsements I've seen, although again I haven't actually read that reference) there are a very readable series of papers by Athel Cornish-Bowden and collaborators (especially a Letelier) where they try to explictly model an (M,R) system in the language of enzymology, to facilitate some hypothesis formulation/testing. This one seems germane: "Closure to efficient causation, computability and artificial life." Journal of Theoretical Biology 263:79–92 (2010)

Mathematics can be used for modeling, any modeling, of anything. Rosen's argument concerning complex systems (natural systems could be subsets of those) states that convoluted (complex) system cannot be characterized by a single (largest) model. So Rosen's arguments concern the adequacy of models of complex systems rather than usability of mathematics for modeling.

I must therefore ask, what is the difference, between mathematical usability and the incompleteness of a mathematical model, especially in terms of utility of resultant model and limitation upon interpretation of obtained result from any suitable inquiry?

It seems to me that incomplete / undecidable mathematics can be and usually is perfectly usable (for instance arithmetic with addition and multiplication is undecidable but usable). So, usability of a mathematical model should not depend of completeness of mathematics involved.

As to complex systems, they should be closed to efficient causation (according to Rosen) and some of the models that represent them can/should be non-simulable/non-computable.

It appears as if there exists a lambda calculus that can handle closure to efficient causation. That is, closure to efficient causation is in principle computable/simulable. Thereby non-simulability of models of complex systems appears to be independent of the closure to efficient causation. [see M.L. Cardenas et al. / Journal of Theoretical Biology 263 (2010) 79–92 and M. Mossio et al. / Journal of Theoretical Biology 257 (2009) 489–498.]

(physical mechanism = objectivity) = limiting case of Science

Rosen claim the necessity to develop a new physic from biology phenomenon, which is a general, rather than 'peculiar' or 'rare' case in Nature. Similar arguments you can find in Autopoiesis and Biology of Cognition from Humberto Maturana.