Dear folks,
I think there is a bit of confusion here due to an ambiguity in the idea of 
computation. A function is computable for a given input only if it has an 
equivalent Turing machine that halts. A function is a computation if it is 
representable by a Turing machine. (I assume the Church-Turing thesis in both 
cases. However there are lots of Turing machines that do not halt (more than 
that do halt). So it is quite possible for a function that is noncomputable to 
be representable by a Turing machine. Wolfram, for example, is fairly clear on 
this. If you know Rosen's work, the computable cases are what he calls 
synthetic models. The noncomputable cases are what he calls analytic but not 
synthetic models. Krivine showed a long time ago that Newtonian mechanics 
allows noncomputable functions that are nontrivial. This is not surprising, 
really, since it is possible to model any Turing machine with a mechanical 
(colliding spheres, say) system. Interestingly, Turing left some work on 
computer models that are not Turing computable.
In any case, the natural computations (to allow Gordana her sense of this idea) 
need not be computable. These cases are nonreducible in the sense of not 
computable from boundary conditions and the combinatorics of lower level 
interactions.  See my A dynamical account of emergence ( ) 
(Cybernetics and Human Knowing, 15, no 3-4 2008: 75-100), 
for some more detail on the reduction and boundary condtions issue. 
Incidentally, to the best of my knowledge it was Conrad, Michael and Koichiro 
Matsuno (1990). The boundary condition paradox: a limit to the university of 
differential equations. Applied Mathematics and Computation. 37: 67-74 that 
first analyzed the boundary system problem. For some even more rigourous 
detail, also C.A. Hooker's chapter on emergence in  C. A. Hooker, Philosophy of 
Complex Systems. Handbook of the Philosophy of Science, Volume 10. 20011: 
Elsevier pp. 195ff.

Professor John Collier  
Philosophy, University of KwaZulu-Natal
Durban 4041 South Africa
T: +27 (31) 260 3248 / 260 2292
F: +27 (31) 260 3031
email:>>> On 2012/05/15 at 03:35 PM, in message 
<>, "Robert Ulanowicz" 
<> wrote:

Quoting Gordana Dodig-Crnkovic <>:

> 2.       Whatever changes in the states of the physical world there  
> are, we understand them as computation.

Dear Gordana,

I'm not sure I agree here. For much of what transpires in nature (not  
just in the living realm), the metaphor of the dialectic seems more  
appropriate than the computational. As you are probably aware,  
dialectics are not computable, mainly because their boundary value  
statements are combinatorically intractable (sensu Kauffman).

It is important to note that evolution (which, as Chaisson contends,  
applies as well to the history of the cosmos [and even the symmetrical  
laws of force]) is driven by contingencies, not by laws. Laws are  
necessary and they enable, but they cannot entail.


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