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## Re: [ontolog-forum] Ontology and Category Theory

 To: "[ontolog-forum]" "Len Yabloko" Fri, 30 Jan 2009 17:36:05 +0000
 ```John,    (01) Thank you for responding to my naive questions about identity. I realize that there is some disconnect between mathematics and reality. As Einstein said http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html    (02) "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."    (03) However, there was more to my quest for unified notion of identity than simply denial of known limits (or limits of known).    (04) JS> >Len, > >It's important to distinguish the fundamental difference between >the physical world and our mathematical theories and constructions. >The world has many complex aspects, some of which are easier to >represent in mathematical models than others. > >LY> I don't think that mathematics in CT in particular were invented > > to turn ordinary notions into schizophrenic image of it. There has > > to be a level at which identity is ... well identity, at least > > for all involved in conversation. > >The mathematical notion of identity is very simple, and it's >embodied in the notation 'x=y' and the axioms that apply to it: > > 1. Reflexivity: For all x, x=x. > > 2. Symmetry: For all x and y, x=y implies y=x. > > 3. Transitivity: For all x, y, and z, x=y and y=z, implies x=z. > >This is the concept of identity in mathematics, including >category theory. It is indeed very clear and very simple. >    (05) Perhaps I placed too much hope in mathematics (non-mathematicians life myself often assume that mathematical terms are just more precise versions of ordinary ones, forgetting about the price one pays for it). I did look into Category Theory publications little bit deeper than reading a title and summary. Admittedly I never could follow one all the way. But I probably got as much of it as an engineer can possibly get. Even after being descurraged by you and Pat, my impression of CT remains to be as of an attempt to make abstract reasoning in general (not only mathematical form of it) more precise. I believe this objective to be a paramount to making it useful beyond calculation and in the real of reasoning. If I am wrong and CT is not attempting to do that, then some other theory should. And my observation is that neither classical Logic nor Ontology as discipline are adequate to this goal if they can't "nail down' the identity. Again, I not talking about absolute and universal identity (I don't know what it is), but about sufficient level of identification required for business transactions.    (06) >The difficulty results from complexity in the world. For example, >is Len at age 2 "identical" with Len at age 22. From the point of >view of many physical attributes, the answer is no. From others, >such as DNA and fingerprints, the answer is yes. Which point of >view is relevant for a particular task will determine which >mathematical theory you apply.    (07) The point of view I prefer is fully expressed in Chris Partridge book, which was mentioned here few times. That is purely extensional approach to reasoning about identity. So in the example of Len at age 2 vs. Len at age 22 I would say they are identical for as long as that fact does not make identities of other objects within scope of transaction(separately of in aggregate). I thought that by defining a category (in CT sense) for all extensions in given scope, in such as way that their identity remains intact, would be a proper application of mathematical abstraction.    (08) > >There is no "ordinary theory" of identity that can make a >decision of which aspects are relevant to any given problem. >    (09) Can we settle on extension being the only relevant aspect?    (010) >No mathematical theory, category theory or anything else, is magic. >You have to decide which aspects of the world to represent in the >mathematics: your size and weight or your DNA and fingerprints.    (011) I don't see a big difference between what mathematicians and magicians do - it is all matter of talent and imagination.    (012) > >John Sowa >    (013) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (014) ```
 Current Thread [ontolog-forum] Ontology and Category Theory, Len Yabloko Re: [ontolog-forum] Ontology and Category Theory, Mitch Harris Re: [ontolog-forum] Ontology and Category Theory, Jakub Kotowski Re: [ontolog-forum] Ontology and Category Theory, Ed Barkmeyer Re: [ontolog-forum] Ontology and Category Theory, Len Yabloko Re: [ontolog-forum] Ontology and Category Theory, Pat Hayes Re: [ontolog-forum] Ontology and Category Theory, John F. Sowa Re: [ontolog-forum] Ontology and Category Theory, Pat Hayes Re: [ontolog-forum] Ontology and Category Theory, John F. Sowa Re: [ontolog-forum] Ontology and Category Theory, Len Yabloko <= Re: [ontolog-forum] Ontology and Category Theory, Mitch Harris Re: [ontolog-forum] Ontology and Category Theory, Pat Hayes Re: [ontolog-forum] Ontology and Category Theory, Len Yabloko Re: [ontolog-forum] Ontology and Category Theory, Mitch Harris Re: [ontolog-forum] Ontology and Category Theory, Pat Hayes Re: [ontolog-forum] Ontology and Category Theory, FERENC KOVACS Re: [ontolog-forum] Ontology and Category Theory, Pat Hayes Re: [ontolog-forum] Ontology and Category Theory, FERENC KOVACS Re: [ontolog-forum] Ontology and Category Theory, Len Yabloko