Thanks for your email Peter. I will indeed followup as you suggest in the next day or two so that readers will be well-informed about these exciting new opportunities for category theory. Best, Joe Peter Selinger <selinger@dead.stanford.edu> writes:
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
-- Joseph R. Kiniry http://www.cs.caltech.edu/~kiniry/ California Institute of Technology ID 78860581 ICQ 4344804
it's interesting that almost no one took michael healy's bait (attached). a couple of years ago, a similar question started a long battle between a group of categorists from this list, and a group of set theorists on another list. traces of that battle can still be found scattered on the web. i guess people got a bit tired. after all, for a working mathematician, foundations are a bit like esperanto: ok, all math can be translated to set theory, or to category theory, or to untyped lambda calculus --- so what? do foundations help me calculate an integral? does anyone use von neumann representation of numbers in arithmetic? no. but guys, note that the question this time comes from cs.boeing.com! if people at boeing think about categories, and sets, and foundations, then this probably makes a difference for them, and helps them compute something. might this not be worth your attention? so let me say what i think. i think foundations mean different things for different people. for logicians, foundations are metamathematics: analyzing consistency, independance etc of logical theories. this is what set theory was found to be good for. category theory, on the other hand, is not as handy for proving new independence results, but it tells you to look for adjunctions everywhere, or monads, or <the keyword from your last paper>. it is not metamathematics, but perhaps *structuralist maths*: it displays abstract structures... (i kno, this is getting to *just* the kind of philosophy you were hoping to avoid by the synchronized silence. so let me make the point.) for software engineers, foundations are the link with the meaning of their programs. having a slightly shorter history than math, they do not have languages as natural as arithmetic, or calculus, but have to chose between KIF and Ontolingua, and the various other versions of esperanto every day. categories dam the flood of structure in software engineering, just like they originally did in homology theory almost 60 years ago. some good math may come out of it if taken from a good side. -- dusko "Michael J. Healy 425-865-3123" 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. --
=========================================================================== 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
============================================================================
I'd like to respond to Dusko's note with an open message to all; I'll try to be brief. I'm grateful to Dusko for his posting, and also to the people who responded to me privately as I'd requested. My only reason for requesting private replies was to avoid any intrusion people might feel if I started another thread such as the "battle" to which Dusko referred:
it's interesting that almost no one took michael healy's bait (attached). a couple of years ago, a similar question started a long battle between a group of categorists from this list, and a group of set theorists on another list. traces of that battle can still be found scattered on the web.
I mean to send a compilation of the responses I received to all respondents because they've all expressed interest, either explicitly or implicitly by responding. I've been short of time and haven't done this yet, partly because I also want to ask each individually if it's OK to use his/her name attached to the response (otherwise, said response will be included as "anonymous"). I will get to it within the next few weeks. In the meantime, I've enjoyed this as a learning experience! Thank you, Mike -- =========================================================================== 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 ============================================================================
Dusko says:
it's interesting that almost no one took michael healy's bait (attached).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> There's a minefield of hotly held opinions behind this question, but let me
Here's a message I sent to Michael Healey privately: try to give a couple of pragmatic considerations. 1. What is the most appropriate characterization (for present purposes) of collections? By elements or by functions? Set theory postulates that a collection is entirely determined by its elements. If functions are better, that's beginning to look more like a categorical approach. Examples: Topology - a space is not fully defined by saying what its points are. You only capture continuity when you look at the maps between spaces. Type theory - syntactic terms in general have free variables in them, effectively parameters, and these really correspond to functions rather than simple elements. 2. Do you want to consider a variety of logics? Set theory as such is solidly classical. To relax that you need to consider different formal systems, and then category theory usually provides tools for achieving better presentation independence than more syntactic approaches. <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< In this hurried response to Michael's posting I was trying to get across some sense of how category theory can help you see the wood for the trees - the "better presentation independence" is what Dusko refers to as "structuralist maths".
for software engineers, foundations are the link with the meaning of their programs. having a slightly shorter history than math, they do not have languages as natural as arithmetic, or calculus, but have to chose between KIF and Ontolingua, and the various other versions of esperanto every day. categories dam the flood of structure in software engineering, just like they originally did in homology theory almost 60 years ago. some good math may come out of it if taken from a good side.
I agree. I looked up keywords like SUO (Standard Upper Ontology) and KIF (Knowledge Interchange Format) - they are about standard notations for expressing information - and some key issues seemed to be rather basic things like the meaning of first order logic. I found this depressing. As we know, categorical logic has some very clear accounts of this that have the great virtue of bringing out deep structure over superficialities of the logical syntax. It also makes it easier to question whether "self-evident" notation and axioms are actually meaningful and valid. But how can we bring these insights to the software engineering masses? It's not enough to say "first learn about category theory and then look for adjunctions, monads and <favourite keyword> everywhere". With this approach it seems all too easy to prescribe people categorical logic as a cure for their myopia, and then to find them trying to use it as reading glasses and wondering why it doesn't work. One of the things that makes Mac Lane's book such a masterpiece is the way it uncovers category theory as something already understood rather than presenting it as something new. The working mathematician is well familiar with reasoning about adjunctions and monads in special cases, and it's "just" a matter of uncovering the underlying pattern. Speaking for myself, after all these years I still understand a lot of category theory not in the abstract but through paradigms. Enriched categories? They're just rings, really. Or, at least, ringoids. Presheaves? They're just modules. Yoneda embedding? The free module on one generator is just the ring itself. The software engineer does not have the working mathematician's body of knowledge. I think to give them category theory we first have to explain, without mentioning categories, just why structure has to reside not inside the objects but amongst the morphisms: to explain a collection by its elements alone is not enough, even in a very basic logical framework. Steve Vickers Department of Pure Maths Faculty of Maths and Computing The Open University ----------- Tel: 01908-653144 Fax: 01908-652140 Web: http://mcs.open.ac.uk/sjv22
participants (4)
-
Dusko Pavlovic -
kiniry＠cs.caltech.edu -
mjhealy＠redwood.rt.cs.boeing.com -
S.J.Vickers＠open.ac.uk