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Pat
Conceptually, a point is a zero
dimensional subspace of an n dimensional manifold (n > 0), In linear algebra,
this means that all the entries of a tuple in the space are set to constants.
There's the rub. No sane mathematician would say, for example, that the vector
(0, 0) is identical to the vector (0,0,0). This raise the question, "what
do we mean by conceptually"
It would appear that when I say,
" Conceptually, a point is a zero dimensional subspace of an n dimensional
manifold " that what I mean is that I can group an infinite family thingies
by a common characteristic, that is, there is an encompassing manifold, and that
within the subspace of the manifold defined by the point thingy, there is no
freedom to move anywhere within that subspace. One might remark that calling
something a point is the opening move in "a game played this way" (I think I'm
quoting Wittgenstein in his Remarks on the Foundations of Mathematics, but my
copy's gone AWOL), and that the conceptulisation of a point refers to a process
of interpretation (of playing a game), rather than saying there is a *thing*
which is "concept of a point". There is no way to evaluate the propstion "X and
Y are the same point", we can only evaluate propositions of the form "In
manifold M, and Y are the same point". Hence we cannot ask "Is
(0, 0) the same point as (0, 0, 0)" since there is no comon
manifold.
So, over the manifold of propositions, TRUE is a single
point. BTW, within the definition of point, there is no specification of the
type of the tuples. If calling TRUE a point seems odd, then I would simply note
that most logics are homomorphic to one of the modulo groups, with classical
logic being mapped to modulo 2. In modulo 2, 2 = 0, and so there is
very little interesting that can be said about the geometry of logic, and so
very little that can usefully be said about TRUE being a
point.
With respect to lines of zero length - they would be
useful in a formal system consisting only of lines to give closure to
operations, on the other hand, if the formal system has both lines and
points,whether zero length lines are an algebraic necessity would depend on the
definition of the operators. BTW, a concept of a type is basically a domain of
values and a set of operators on the valures. Mathematicians generally like
operators that are total (apply to every value or value tuple) and which the
result is also a value in the domain. Hence the remark on
closure.
Sean Barker
*** WARNING ***
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Sean,
I could use some clarification:
[SB]
> For
example, I would regard 2D and 3D points as referring to different concepts,
whereas Cartesian co-ordinate systems v. polar co-ordinate systems for a 2D
point as different reifications of the same
concept.
I have
been assuming that a point has zero dimensions, and can exist in any coordinate
system of any dimensionality. There is an issue as to whether, for
example, a 1D line of zero length is identical to a point. That may
require a translation. What have I missed?
Pat
Patrick
Cassidy
MICRA,
Inc.
908-561-3416
cell:
908-565-4053
cassidy@xxxxxxxxx
From:
ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of sean
barker
Sent: Wednesday, March 10, 2010 2:34 PM
To:
ontolog-forum@xxxxxxxxxxxxxxxx
Subject: Re: [ontolog-forum] Re
Foundation ontology, CYC, and Mapping
Duane
Chris
Menzel was right in saying it's not that subtle. The term type here refers to
the abstract data type used to reify the concept - for example, does one reify a
latitude as a real number, a fixed precision decimal number, or as a triple of
integers for degree, minute, second.
This is
distinct from the problem 3D v 4Dism that Matthew referred
to.
One of
the problems I have not seen discussed much - possibly because I have been
looking in the wrong place - is the relation in ontology
languages between concepts and their reification, as opposed to the
relation between different concepts. For example, I would regard 2D and 3D
points as referring to different concepts, whereas Cartesian co-ordinate systems
v. polar co-ordinate systems for a 2D point as different reifications
of the same concept. Looking at languages like OWL, it seems that the
reification is identified with form of the concept, as if there
is only one way of reifying it.
Having
two different reifications of a concept should not be a major semantic
challenge, the challenge is that, unless you account for the different
reifications, the systems cannot interoperate. However there may be practical
problems concerning the adequacy of the reifications. See, for example, Cliff B
Jones, "Systematic Software Development using VDM", Chapter 8 on Data
reification for a more detailed treatment.
The
converse is what John Sowa keeps insisting on, that interoperation happens
mostly at the level of middle ontologies. In this case, there is some morphism
between the reifications - or at least a subset of the reifications - which can
be used for interoperation. For example, there is a simple morphism between
points in Euclidean space and those in a homogenious co-ordinate system. In one
dimension this is
E(x) -> H(x, 1) and H(x, 1) -> E(x).
This
breaks down for points of the form H(x, 0), but then Eucllidean spaces doesn't
have a lot to say about points at infinity.
Sean Barker
Bristol
From:
ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Duane
Nickull
Sent: 09 March 2010 22:41
To:
[ontolog-forum]
Subject: Re: [ontolog-forum] Re Foundation ontology,
CYC, and Mapping *** WARNING ***
This message has originated outside your organisation,
either from an external partner or the Global Internet.
Keep this in mind if you answer this message.
Sean:
For
the second of these (conflicts when the same concept is represented by different
types), can you elaborate a couple of examples (no hurry). I just want to
make sure I have a good idea of this.
Duane
On 3/9/10 2:30
PM, "sean barker" <sean.barker@xxxxxxxxxxxxx>
wrote:
Apologies for slow
response to a couple of requests for sources on semantic
incompatibilities.
This is the table we generated internally, based partly on
older database work
Semantic
Incompatibilities
Naming
Conflicts When objects
representing the same concept may contain dissimilar names: conflicts due
to either homonyms or synonyms.
Type
Conflicts When the same
concept is represented by different types.
Key
Conflicts When different
keys are assigned to the same concept in different schema.
Behavioural
Conflicts When different
insertion/deletion policies are associated with the same class of objects
in different schemata. e.g. deleting an object may leave an ?empty? object
rather than a ?null reference?.
Missing
Data When different
attributes are defined for the same concept.
Levels of
Abstraction When
information about an entity is stored at dissimilar levels of detail. e.g.
?name? versus ?first_name? and ?last_name?.
Identification of
Related Concepts For example,
two entities belonging to two different databases may not be equivalent
but one entity may be a generalisation of the other
entity.
Scaling
Conflicts When the same
attribute of an entity is stored in dissimilar
units.
it is based on/taken
from
[1]
Aykut Firat,
Information Integration Using Contextual Knowledge and Ontology Merging.
MIT (Sloan School of Management) Ph. D thesis, September
2003.
[1]
M. P. Reddy, B. E.
Prasad, P. G. Reddy, Amar Gupta, A Methodology for Integration of
Heterogeneous Databases, IEEE Transactions on Knowledge and Data
Engineering, Vol. 6, No. 6, December 1994.
There are some other papers dating
from the mid-nineties, but they have not survived my various office moves.
Sean
Barker
Bristol
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