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Re: [ontolog-forum] Ontology and Category Theory

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 30 Jan 2009 11:04:43 -0500
Message-id: <4983251B.3060305@xxxxxxxxxxx>
Len,    (01)

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.    (02)

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.    (03)

The mathematical notion of identity is very simple, and it's
embodied in the notation 'x=y' and the axioms that apply to it:    (04)

  1. Reflexivity:  For all x, x=x.    (05)

  2. Symmetry:  For all x and y, x=y implies y=x.    (06)

  3. Transitivity:  For all x, y, and z, x=y and y=z, implies x=z.    (07)

This is the concept of identity in mathematics, including
category theory.  It is indeed very clear and very simple.    (08)

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.    (09)

There is no "ordinary theory" of identity that can make a
decision of which aspects are relevant to any given problem.    (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)

John Sowa    (012)

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