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## [ontolog-forum] Fw: Re Foundation ontology, CYC, and Mapping

 To: "sean barker" Thu, 11 Mar 2010 19:47:17 -0000 <080EDC85D08A4A7E95070ABFD7705F62@SMB>
 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   From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Patrick CassidySent: 10 March 2010 19:50To: '[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,    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 barkerSent: Wednesday, March 10, 2010 2:34 PMTo: ontolog-forum@xxxxxxxxxxxxxxxxSubject: 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 BarkerBristol From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Duane NickullSent: 09 March 2010 22:41To: [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. DuaneOn 3/9/10 2:30 PM, "sean barker" 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 BarkerBristol
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