I'd like to ask category theorists how they would answer the attached message from a colleague here. Both he and the person with whom he is corresponding are experts in the areas of knowledge representation within computer science (ontologies and the like). I thought it best to hide their identities since I haven't asked permission to use them. If you are interested, please respond to me privately if you would. Thank you, Mike Healy ------------------------------------------------------------------------------ Message I received:---- I would be delighted if there was no semantic conflict between category theory and set theory. I kind of flagged this as a potential issue, but did not look into it in detail, as it was not my main concern at the time. However, I remain unconvinced. There has been some discussion of using set theory as the basis for a semantics for SUOKIF. If this is true, then I think it may be limiting to a CT based language. While it may be true that sets are common example of a catagory, my sense is that CT is much more powerful, and would be LIMITED if everything was forced into the single catagory of sets. Im a bit out of my element here, however, and need to defer to the formal expertise of others on this issue. Message to which the above was replying:--- I agree that category theory is very powerful and could be an important basis for combining and sharing ontologies. But I disagree with the following point:
I think this idea has tremendous potential. One problem is that the underlying
formal semantics of category theory is NOT set theory (which is what KIF uses),
furthermore, I think they may well be incompatible.
First-order logic (including any and all notations for it, such as KIF, CGs, predicate calculus, existential graphs, etc.) is completely neutral with respect to set theory or category theory. The version 3.0 of KIF did include a version of set theory, but that was removed in the KIF'99 version because it belongs to ontology rather than logic. And for that matter, there is no reason why you can't use both category theory and set theory together. In fact, one of the most common examples of a category is the category of sets. Perhaps there may be incompatibilities between the methodology associated with Ontolingua and category-based techiques, but Ontolingua is not KIF. Ontolingua simply uses KIF. -- =========================================================================== e Michael J. Healy A FA ----------> GA (425)865-3123 | | FAX(425)865-2964 | | Ff | | Gf c/o The Boeing Company | | PO Box 3707 MS 7L-66 \|/ \|/ Seattle, WA 98124-2207 ' ' USA FB ----------> GB -or for priority mail- e "I'm a natural man." 2760 160th Ave SE MS 7L-66 B Bellevue, WA 98008 USA michael.j.healy@boeing.com -or- mjhealy@u.washington.edu ============================================================================
Hello Michael, I believe that category theory is an excellent foundation for ontology representation and manipulation -- I use it myself. However, choosing this foundation comes at a price. Unless significant work is done to hide this unfamiliar foundation, many users (of the theory, the system, the language, &c) will be biased against the work from minute-one. This has as much to do with the unfamiliarity of CT as it does with certain unfortunate negative biases regularly expressed by many mathematicians and computer scientists - biases that, IMHO, are founded in ignorance and not reason. Many computer scientists, mathematician, and users of knowledge representation systems are quite familiar (at least in use, but probably not in foundations or related complications) and comfortable with set theory. This familiarity is an incentive rather than an obstacle to using related work. Personally, I chose not to pursue a set theoretical foundation because of theoretical and representational complexity issues (e.g. witness the use of a set theoretical foundation for the Z specification language) as well as the unfortunate binding to a particular formalism that isn't necessarily congruent with others that I work in and apply my work to (e.g. type theory and programming languages). To rephrase, I find using CT to be more clear and tractable than set theory and I feel that my work can, as a result, say and do more than it could if it had a set theory (plus some extra formalisms) bases. Note that I also chose not to build my work (solely) on type theory and order sorted algebras for the same reason, though my work has elements of both of those fields as well. The comments about Ontolingua and KIF are on-target in my experience. I see no obstructions to the representation of CKML (the variant related to KIF and RDF that I happen to know well) with my work. Finally, I should point out that I am but an infant in CT - I'm much more comfortable with TT, OSA, PL, and others. I've only been learning and using CT for a few months and, while there have been some objection to my choice, I feel that a dissertation founded in these three major fields (CT, TT, and OSA) has significantly broader application and, implicitly, more to say about its author. <grin> Best, Joe Kiniry -- Joseph R. Kiniry http://www.cs.caltech.edu/~kiniry/ California Institute of Technology ID 78860581 ICQ 4344804 --On Tuesday, January 16, 2001 04:17:19 PM -0800 "Michael J. Healy 425-865-3123" <mjhealy@redwood.rt.cs.boeing.com> wrote:
I'd like to ask category theorists how they would answer the attached message from a colleague here. Both he and the person with whom he is corresponding are experts in the areas of knowledge representation within computer science (ontologies and the like). I thought it best to hide their identities since I haven't asked permission to use them. If you are interested, please respond to me privately if you would.
Thank you, Mike Healy ------------------------------------------------------------------------- -----
Message I received:----
I would be delighted if there was no semantic conflict between category theory and set theory. I kind of flagged this as a potential issue, but did not look into it in detail, as it was not my main concern at the time. However, I remain unconvinced. There has been some discussion of using set theory as the basis for a semantics for SUOKIF. If this is true, then I think it may be limiting to a CT based language. While it may be true that sets are common example of a catagory, my sense is that CT is much more powerful, and would be LIMITED if everything was forced into the single catagory of sets.
Im a bit out of my element here, however, and need to defer to the formal expertise of others on this issue.
Message to which the above was replying:---
I agree that category theory is very powerful and could be an important basis for combining and sharing ontologies. But I disagree with the following point:
I think this idea has tremendous potential. One problem is that the underlying
formal semantics of category theory is NOT set theory (which is what KIF uses),
furthermore, I think they may well be incompatible.
First-order logic (including any and all notations for it, such as KIF, CGs, predicate calculus, existential graphs, etc.) is completely neutral with respect to set theory or category theory. The version 3.0 of KIF did include a version of set theory, but that was removed in the KIF'99 version because it belongs to ontology rather than logic.
And for that matter, there is no reason why you can't use both category theory and set theory together. In fact, one of the most common examples of a category is the category of sets.
Perhaps there may be incompatibilities between the methodology associated with Ontolingua and category-based techiques, but Ontolingua is not KIF. Ontolingua simply uses KIF. --
Michael J. Healy
Dear Joseph, I don't presume to speak for everybody on the mailing list, but it is probably safe to say that the majority of readers have no clue what Z, Ontolingua, KIF, CKML, RDF, TT, OSA, or PL are. Certainly I don't. If this is an exciting area of applications of category theory, then category theorists should know about it. If you have time, I think it would be great if you could briefly explain these terms to the list, and maybe say a word or two specifically about how category theory is used there. Thanks, -- Peter Selinger Joseph R. Kiniry wrote:
Hello Michael,
I believe that category theory is an excellent foundation for ontology representation and manipulation -- I use it myself.
However, choosing this foundation comes at a price. Unless significant work is done to hide this unfamiliar foundation, many users (of the theory, the system, the language, &c) will be biased against the work from minute-one. This has as much to do with the unfamiliarity of CT as it does with certain unfortunate negative biases regularly expressed by many mathematicians and computer scientists - biases that, IMHO, are founded in ignorance and not reason.
Many computer scientists, mathematician, and users of knowledge representation systems are quite familiar (at least in use, but probably not in foundations or related complications) and comfortable with set theory. This familiarity is an incentive rather than an obstacle to using related work.
Personally, I chose not to pursue a set theoretical foundation because of theoretical and representational complexity issues (e.g. witness the use of a set theoretical foundation for the Z specification language) as well as the unfortunate binding to a particular formalism that isn't necessarily congruent with others that I work in and apply my work to (e.g. type theory and programming languages). To rephrase, I find using CT to be more clear and tractable than set theory and I feel that my work can, as a result, say and do more than it could if it had a set theory (plus some extra formalisms) bases.
Note that I also chose not to build my work (solely) on type theory and order sorted algebras for the same reason, though my work has elements of both of those fields as well.
The comments about Ontolingua and KIF are on-target in my experience. I see no obstructions to the representation of CKML (the variant related to KIF and RDF that I happen to know well) with my work.
Finally, I should point out that I am but an infant in CT - I'm much more comfortable with TT, OSA, PL, and others. I've only been learning and using CT for a few months and, while there have been some objection to my choice, I feel that a dissertation founded in these three major fields (CT, TT, and OSA) has significantly broader application and, implicitly, more to say about its author. <grin>
Best, Joe Kiniry -- Joseph R. Kiniry http://www.cs.caltech.edu/~kiniry/ California Institute of Technology ID 78860581 ICQ 4344804
--On Tuesday, January 16, 2001 04:17:19 PM -0800 "Michael J. Healy 425-865-3123" <mjhealy@redwood.rt.cs.boeing.com> wrote:
I'd like to ask category theorists how they would answer the attached message from a colleague here. Both he and the person with whom he is corresponding are experts in the areas of knowledge representation within computer science (ontologies and the like). I thought it best to hide their identities since I haven't asked permission to use them. If you are interested, please respond to me privately if you would.
Thank you, Mike Healy ------------------------------------------------------------------------- -----
Message I received:----
I would be delighted if there was no semantic conflict between category theory and set theory. I kind of flagged this as a potential issue, but did not look into it in detail, as it was not my main concern at the time. However, I remain unconvinced. There has been some discussion of using set theory as the basis for a semantics for SUOKIF. If this is true, then I think it may be limiting to a CT based language. While it may be true that sets are common example of a catagory, my sense is that CT is much more powerful, and would be LIMITED if everything was forced into the single catagory of sets.
Im a bit out of my element here, however, and need to defer to the formal expertise of others on this issue.
Message to which the above was replying:---
I agree that category theory is very powerful and could be an important basis for combining and sharing ontologies. But I disagree with the following point:
I think this idea has tremendous potential. One problem is that the underlying
formal semantics of category theory is NOT set theory (which is what KIF uses),
furthermore, I think they may well be incompatible.
First-order logic (including any and all notations for it, such as KIF, CGs, predicate calculus, existential graphs, etc.) is completely neutral with respect to set theory or category theory. The version 3.0 of KIF did include a version of set theory, but that was removed in the KIF'99 version because it belongs to ontology rather than logic.
And for that matter, there is no reason why you can't use both category theory and set theory together. In fact, one of the most common examples of a category is the category of sets.
Perhaps there may be incompatibilities between the methodology associated with Ontolingua and category-based techiques, but Ontolingua is not KIF. Ontolingua simply uses KIF. --
Michael J. Healy
participants (3)
-
Joseph R. Kiniry -
mjhealy＠redwood.rt.cs.boeing.com -
selinger