Well said! And very useful. Your set of links make this a useful resource for all. I hope that we can explore many of the issues that you have brought here over the next 4 months, both here in this blog, and in class. I appreciate your skepticism, but I, too, have come to believe Paul's assertion (not proof) that some deterministic systems (where there is absolutely no chance of chance entering the picture) cannot be predicted. Whether or not you find a way to predict them in the future, I think that this is the fundamental difference between "computational" models, versus "mathematical" models. And a key aspect of emergence. We'll discuss this more, no doubt. (Also, I don't think that one can eliminate the divine... it isn't science, and so lies outside of that possibility. But that's another thread...)

You've raised an interesting set of issues here, and provided some good links for further exploration of them. Let me suggest it might be useful to disentangle the issues a bit, and provide some additional links that might be relevant along those lines. "skeptical ... that ... emergent processes cannot be predicted at an earlier stage but must played out via a computer simulation ... Who’s to say that there isn’t another branch of mathematics waiting to be discovered that can predict these computer generated reactions?" It is in fact the case that the future state of certain systems of simple interactions among simple things can be shown not to be predictable from the initial conditions using any "formal" mathematical system, ie any mathematical system with a fixed set of axioms, rules of inference from those axioms, and procedures for verifying proofs. The form of demonstration orginates with Godel's "incompleteness theorem", was developed further by Turing with specific reference to computers in connection with the "halting problem", and has implications for what one means by "science" that are still being explored (see Gregory Chaitin's "Omega and why math has no theories of everything", Wolfram on "computational irreducibility", and my own "Science As Story Telling and Story Revising"). The point is that "mathematics", like other things, is continually evolving (emerging?) and so one has to be explicit about what one means by that term at any given time. It may well be possible to develop intuitions (and perhaps even rules/generalizations) about systems that must, at the moment, be played out in simulations; that's part of what we (and others interested in emergence) are exploring. What can be said, for many of them, is that such new intuitions/rules/generalizations will not be in the form of "explicit solutions", ie of equations that yield the state of the system at a future time simply by plugging in the value of time. "I don’t want there to be a science that cannot be governed by mathematical laws, only observed. Admitting that emergence processes cannot be predicted or understood leaves room for the divine. And I don’t find the divine very useful in applied scientific enquiry." "cannot be predicted" is different from cannot be "understood", and even more so from "cannot be usefully inquired into", which seems to me the only thing that would put something outside the reach of science (see Science, Pragmatism, and Multiplism, as well as Science As Story Telling and Story Revising, and Science as Storytelling or Story Telling?). Behe's "irreducible complexity", with its associated interest in establishing the existence of a creator or architect, is quite different from the Godel, Turing, Chaitin interest in what is and is not "computable" in particular ways, which makes no necessary assumption that any limitations described are evidence for a divine creator/architect. Indeed, in my own case (at least) the interest derives entirely from an engagement with the question of how far one can make sense of things in the absence of any presumed creator/architect. Yes, indeed there is, of course, a relation between all of this and the "Intelligent Design versus Darwinism debate" (cf Intelligent Design and the Story of Evolution: No Need for Drawing Lines in the Sand, Fundamentalism and Relativism: Finding a New Direction, and Science as Storytelling or Story Telling?). But it is not a simple relation. Emergence is an inquiry about possibility, not a provable description of the "real world". Moreover, the emergence perspective does not inevitably imply either "scientific determinism" or "falsity of human free will" (any more than limitations on predictability inevitably imply an architect/designer). One might instead, from the emergence perspective, pose the question (as we will) of how both architects/designers and "free will" might emerge from simple interactions among simple things. And conclude (as we might) that some degree of indeterminacy is not only acceptable as a scientific concept but essential to account for what we are all trying to make sense of (cf. Variability in Brain Function and Behavior and Free Will?).