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On Tuesday, January 27, 2009 5:37 PM, Pat wrote:
"The point is that ontologies work by describing, and logically expressed
ontologies are not limited to only describing finite or discrete structures or
phenomena."
That's true. Just to remind some basics from the classic source of the
issue: "Quantity is either discrete or continuous.... Instances of discrete
quantities are number and speech; of continuous, lines, surfaces, solids, and,
besides these, time and place".
----- Original Message -----
Sent: Tuesday, January 27, 2009 5:37
PM
Subject: Re: [ontolog-forum] Ontological
Means for Systems Engineering
On Jan 27, 2009, at 12:23 AM, Azamat wrote:
Interesting. This is a Wikipedia stub, all the actual text of
which is lifted from a cited page in Wolfram Mathworld, which lists the
following topics under 'discrete mathematics':
Cellular Automata (45)
Coding Theory (37)
Combinatorics (13)
Computational Systems (42)
Computer Science (7)
Division Problems (29)
Experimental Mathematics (31)
Finite Groups@
General Discrete Mathematics (5)
Graph Theory (14)
Information Theory@
Packing Problems@
Point Lattices (44)
Recurrence Equations (51)
Umbral Calculus (35)
Note the absence of set theory and logic. My own mathematical education
certainly did not classify set theory as discrete mathematics, and did not
consider logic to be part of mathematics at all. I believe the the anonymous
author of the Wikipedia stub was thinking of the study of logic as a topic
(rather then its use as a formalism) when they listed it here. And of course,
logical syntax and logical proofs are indeed finite, discrete, structures. But
as I said earlier, that does not imply that ontologies written in logics
suffer from a "limitation" regarding their ability to describe continuity
(which is what Jeffrey's email seems to suggest, and to which I was
responding.) The point is that ontologies work by describing, and logically
expressed ontologies are not limited to only describing finite or discrete
structures or phenomena.
Pat Hayes
Discrete mathematics includes the following topics:
----- Original Message -----
Sent: Monday, January 26, 2009 11:42
PM
Subject: Re: [ontolog-forum] Ontological
Means for Systems Engineering
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.
>
> >
>
> > Pat Hayes replied
>
> >
>
> > ??? No they aren't. Where
did you get this odd notion from?
>
> > PatH
>
>
>
> By which I mean such as set theory, algebra, combinatorics, logic,
etc.
>
>
>
> -- Jeff
>
> 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.
>
>
PatH
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.
Pat H.
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