Thank you for responding to my naive questions about identity. I realize that
there is some disconnect between mathematics and reality. As Einstein said
"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.
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.
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)
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