On Jan 26, 2009, at 12:21 PM, Schiffel, Jeffrey A wrote:
> > > > Jeffrey Schiffel originally wrote:
> > > >... A system is smaller than the world. A system of systems > > > > is still very small compared to the world. They each have a > > > > known number of interacting parts. But the nature of the > > > > interactions is perhaps limits how ontologies in the usual > > > > sense can be used. The limitation is that ontologies are based > > > > in discrete mathematics. > > > ??? No they aren't. Where did you get this odd notion from? > > By which I mean such as set theory, algebra, combinatorics, logic, etc. > OK. But set theory is not particularly discrete. Sets are the basis for all mathematics, > including continuous mathematics. Similarly, logic is not particularly concerned with > discrete structures: one can use logics to describe the continuum. (There is a > technical issue in that purely first-order logic cannot completely characterize the > finite/infinite distinction and hence cannot fully describe continuity; but then pure > FOL cannot fully characterize finitude, either.) > On the other hand, ontologies don't seem to have any obvious connection > with combinatorics or algebra.
OK, not “algebra,” but “algebras.” I have in mind, for example, functions and similar transformational mappings, topological spaces, and similar structures that are sometimes brought up in discussions of ontological structures.
Im getting more and more confused. The mathematical term "function" just means any mapping, continuous or discrete. Topology is the abstract study of continuous
spaces, about as far from being discrete mathematics as one can get.
Surely combinatorics and ontologies both involve graph theory, a branch of discrete mathematics. Combinatorics, with the theory of algorithms (another discrete mathematics) is used in ontology applications such as data mining and knowledge bases.
Logical (and more generally, KR) syntax
is of course finite and discrete, like any syntax, and hence operations on it are analyzed using techniques from discrete math, as with almost all other algorithms, but that doesn't imply that ontologies are limited to describing
only discrete topics.
I certainly understand that logics are used in describing the continuum, and have a place in real analysis.
Recall that the original query from Andreas Tolk was the application of ontologies in systems engineering. I pointed out that systems engineering has quite a bit to do with complexity and hierarchy, part of the continuous mathematics.
Complexity theory uses analysis, of course, but hierarchies are discrete structures.
Consider also the useful threads previously that discussed time as an ontology element. This, too, suggests that a continuous component does not easily fit into a generally accepted method of ontology development involving concepts and instantiations of concepts.
?? I completely fail to follow your reasoning here. First, time (as described in ontologies) can be continuous or discrete: both kinds of temporal models have their uses and limitations. So why do all these extended discussions about time (continuants vs. occurrents, etc.) suggest anything at all about continuity? All these issues and debates arise whether time is conceptualized as continuous or as discrete (number of milliseconds since 1960, etc.).
My experience in extremely large systems development has taught me that the hierarchy of systems, subsystems, and primitives is too often minimized, leading to failures during development and deployment.
That seems quite plausible, but again seems not to have anything to do with continuous mathematics.
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