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 ( http://web.ncf.ca/collier/papers/A%20Dynamical%20Account%20of%20Emergence.pdf ) (Cybernetics and Human Knowing, 15, no 3-4 2008: 75-100), http://web.ncf.ca/collier/papers/A%20Dynamical%20Account%20of%20Emergence.pdf 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. Cheers, John
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: colli...@ukzn.ac.za>>> On 2012/05/15 at 03:35 PM, in message <20120515093552.322364lbu120x...@www.cbl.umces.edu>, "Robert Ulanowicz" <u...@umces.edu> wrote: Quoting Gordana Dodig-Crnkovic <gordana.dodig-crnko...@mdh.se>: > 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. Regards, Bob _______________________________________________ fis mailing list fis@listas.unizar.es https://webmail.unizar.es/cgi-bin/mailman/listinfo/fis Please find our Email Disclaimer here: http://www.ukzn.ac.za/disclaimer/
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