Hello, I keep being surprised by this list, when topics which I have been interested in for some time come cropping up in the discussion here :-)
I would like to second Robert's recommendation for the book Conceptual Mathematics, a good place to go from there would be: Category Theory http://www.oup.com/uk/catalogue/?ci=9780198568612 from Steve Awodey: http://www.andrew.cmu.edu/user/awodey/ I haven't read the latter yet (I'm still working throught Conceptual Math.)but this is definitely the book I will go for next - Awodey has a good reputation for being accessible without making compromises regarding content (the book is rather expensive unfortunately). The classic on Category Theory is Mac Lane: Categories for the Working Mathematician. but, as the title suggests, is for fully-fledged mathematicians only ;-) Regards, Günther steve smith wrote: > Owen, et alii - > >> >> Has anyone used this in complexity science work? Or semantic web >> work? Or anything else? :) > > > My colleagues at UNM... Tom Caudell (whom I believe you have met), Tim > Goldsmith (Cognitive Psychologist) and Mike Healey (Mathematician > retired from UWash) are using it to develop knowledge models from expert > elicitation. The methodology they are developing is essentially > (apologies to them for any mistakes I make) as follows: > > 1) Collect a set of potential "experts". > 2) Interview them about the topic in question, primarily asking what > words (terms) they use ot describe the topic, think about the topic, > pontificate on the topic. > 3) Pile all these terms on a big blanket out in the field on a windy day. > 4) Toss the terms in the air and let the wind carry away the lightweight > and trivial ones. > 5) Sort through the remainders and join up synonyms . > 6) Go back to the experts and ask them to rank the pairwise distance > between terms. (N squared!) One gets a fully connected graph. > 7) Do some kind of normalization thingy amongst the results... call it a > numerical average for now. > 8) Threshold the edges such that the graph no longer is fully connected > (black magic mojo). > 9) Iteratively consult a subset ( the more cooperative ones?) of experts > on steps 7, 8. > 10) Viola! > > Although I am only peripherally involved in their discussions on this, I > believe: > A) 8) There are probably more advanced graph theoretic things to do than > simply threshold the weights... like collapsing cycles and/or finding > some heirarchy, and/or thresholding some more interesting??? derived > measure than the simple, original weights... maybe... > B) 7) There are likely somewhat interesting things to do here, > especially to (later) place the different experts "point of view" > relative to the collective. There would seem to be a lot of soft > and/or unknown factors regarding the nature of the experts... etc. > > I'm trying to converge my own less formal theories about Metaphor in > Information Visualization (formal analogy, etc) with their work, but > there is still a bit of distance (probably entirely in my lack of > understanding of the nuances of category theory). My now-30-year old > BS in Mathematics and Physics with a handful of graduate courses in > group theory and topology tossed on top for garnish serves me just well > enough to get in trouble... > > I have been doing work in Visualization of Ontologies which also seems > to relate... I'm not sure anyone knows how to build an ontology > really... or how to describe the caveats and conditions surrounding the > Ontology. The Gene Ontology I have worked most with seems to have > plenty of anomolies of both history and of the compromises made to bring > it to a single, agreed-upon ontology... > > It seems that most Ontologies, at least for the moment are going to be > self-organizing... that the only people both able and willing to build > such a huge abstract beast are those who will also use it... > > One problem (in my opinion) is that it is somewhat of a "theory of > everything" so in some sense, all formal knowledge models can be > expressed in or traced back to category theory... so merely saying that > one is "using category theory" is not unlike replying to the question > "How did you get here?" with "I used a mode of transportation". > > For example, at a meeting between Caudell and two of my more strongly > mathematically inclined colleagues last week, it was stated with > complete confidence and agreement around the the table that Formal > Concept Analysis is "just a specific use of Category Theory"... > > >> We've knocked around the term Category Theory a bit lately, so I >> started looking into it a bit. This seems to be a reasonable >> starting place: >> http://en.wikipedia.org/wiki/Category_theory > > Wikipedia strikes again! I am constantly amazed at how accessible and > thorough technical information on Wikipedia is. I can't vouch for it's > accuracy (or thoroughness) in this case, but I am impressed at how well > these articles seem to summarize what I think I already know and plenty > I'm still trying to figure out. > > And to make it even more interesting... isn't Wikipedia a > self-organizing ontology of everything? If one "labels" the links used > in Wikipedia to other Wikipedia elements with the verbs used in the > text, does that not begin to make an ontology? > > Like the first line In Categories: > > mathematics, categories allow one to formalize notions involving > abstract structure and processes that preserve structure. > > We have a link between "Categories" and "Mathematics" and perhaps > (suggesting new links or topic are needed in Wikipedia) "Notions" and > "Structure" or perhaps "Abstract Structure" and "Processes", etc.. > > I look forward to the evolution of this discussion here (If I can even > keep up). > > - Steve > > > ------------------------------------------------------------------------ > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
