[Top] [All Lists]

Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Len Yabloko" <lenya@xxxxxxxxxxxxx>
Date: Fri, 30 Jan 2009 17:36:05 +0000
Message-id: <W3457318039113011233336965@webmail22>
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)

>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)

<Prev in Thread] Current Thread [Next in Thread>