Günther, I have admitted my ignorance of your domain: a philosophy of science. Apparently, this knowledge is required in FRIAM discussions. Perhaps there is a path between the philosophy behind of, and the science I use in my complexity research.
A hint of this path is illustrated in Robert Bishop and Harald Altspacher's "Contextual Emergence in the Description of Properties". http://philsci-archive.pitt.edu/archive/00002934/ Among other issues of description, they state: "The description of properties at a particular level of description (including its laws) offers both necessary and sufficient conditions to rigorously derive the description of properties at a higher level. This is the strictest possible form of reduction. As mentioned above, it was most popular under the influence of positivist thinking in the mid-20th century." The positivist paradigm is inadequate in my opinion. Models can be represented by hybrids of thermodynamic network graphs and neural networks. The point I tried to make in Gödel is that models that are incomplete may be proven, what I call true in some sense is actually more not-false, and models that are unproven may be true. Models are superpositions of other models that may be refuted as demonstrably false and removed from the superposition, otherwise allowed to remain. Therefore, models exhibit uncertainty. Axioms are not models, and models cannot represent axioms. If a model is complete, the ensembles of entangled paths through the superposition will collapse to a trajectory. In my domain, a model may be considered in two types of equilibrium, both where no energy is exchanged with the context. 1) A cold dark model, thermodynamically just above 0 K where no thermodynamic coupling exists between elements - thus entropy is not generated), and 2) a dark model where the model is at thermodynamic equilibrium with its environmental context, but exchanges internal and external couplings. A dark model does generate entropy at its minimal level according to various ambient temperatures. The upshot of the general description above is that perturbations of the model can realize non-analytical solutions (of which there may be infinitely many) - which are impossible with, and completely different than, solving a problem analytically. Re: completeness your propositional calculus, I am ignorant of the concept. Ken > -----Original Message----- > From: Günther Greindl [mailto:[EMAIL PROTECTED] > Sent: Thursday, July 17, 2008 12:05 PM > To: [EMAIL PROTECTED]; The Friday Morning Applied > Complexity Coffee Group > Subject: Re: [FRIAM] Confessions of a Mathemechanic. > > Ken, > > > proven as true in a formal axiomatic system. Thus, "truth" is an > > underdetermined state when it comes to the application of enumerable > > It is always important to say here that "truth" in respect to > Gödel is a mathematical notion (relationship structure/model > and formal system), it is often wrongly invoked in > philosophical discussion ("Gödel said there can be no truth > .. therefor crazy idea etc") > > > > Gödel's second theorem states that a formal axiomatic system is > > complete if and only if it is inconsistent. > > There are perfectly complete and and consistent axiomatic systems. > (propositional calculus); heck, even the mega-expressive > first order logic (see the completeness theorem). > http://en.wikipedia.org/wiki/Completeness_theorem > > Incompleteness arises when you introduce arithmetic (robinson > arithmetic suffices, presburger arithmetic not; in short: you > need addition and multiplication in your arithmetic -> with > this you can construct gödel numbers, define recursion, and > get your (first) incompleteness theorem, from which second > follows easily. > > Cheers, > Günther > > -- > Günther Greindl > Department of Philosophy of Science > University of Vienna > [EMAIL PROTECTED] > > Blog: http://www.complexitystudies.org/ > Thesis: http://www.complexitystudies.org/proposal/ ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
