Re: set theory and categories

[Judith Rosen <***>]

I generally don't get involved in the mathematics when such things are
discussed in specific terms-- mainly because the specifics were not of
interest to me, personally, and I therefore never took the time to learn the
specifics (as in the fine points of differential equations as a modeling
tool for blah, blah, blah, etc). However, this is becoming a general
discussion, and the generalities ARE of interest. When it comes to set
theory as a mode of notation, or a language, I've discussed these aspects
with my father and I know far more about them than I do the specific math
issues.

Set theory is a huge area that can subsume category theory in the following
way: in terms of  "sets of categories" and "categories of sets" and so on.
My father never abandoned set theory in his work. He did get more specific
in how he applied certain concepts and perhaps that is what Ionel is
reacting to. I'm sure my father's response, in a general way, to this
discussion would be that there is not a "cut" between sets and categories.
(Although there may well be a "cut" between "Set Theory" (note the capitals)
and "Category Theory" as defined by individual Mathematicians... just as
there is a difference between Rosennean Complexity Theory and other
complexity theories.)

Robert Rosen could, of course, speak much more at length to the specifics
here! Since I think it might be useful to the list, I'll mention that I have
a "math ace", as I call him, a former PhD student of my father's, who can
speak to the mathematical specifics, and I will forward this discussion on
to him and ask him to give me his "take" on it. I'll post the response he
gives me to the list. However, my (general) knowledge of my father's reasons
for using set theory is fairly well developed:

Set theory, as my father used the term in our discussions, referred to a
language that allowed a visual and non-prose mode of notation of ideas,
which referred to different ways of defining "things". Depending on how such
things were defined, they were seen as being part of various sets, which
could then be shown, via the same notation, to be overlapping in such ways
as these same things were obviously also members of innumerable other sets.
That means that such members of any set which was also a member of other,
overlapping, sets, ("common subsets") was a "thing" with multiple
definitions. In this way, you can use set theory notation to illustrate the
fact that any part of an organism, for example, is also something else to
that organism, or even infinite "something else"s. Complexity depends
entirely on the multidimensionality that results from the organization
having an active role in creating new "sets" and new "subsets". Relational
issues arise even in complex systems that are not biological, like the atom.
But biological systems-- living systems-- have a new "set" of properties:
life, function, anticipation, etc, which put the sets of sets of sets into
categories (more for ease of expression than anything else, it seems to me).
So, in this way of mathematically illustrating the relational nature of
organization in living systems, and the word "categories" naturally fits
into this framework.

The danger, in using mathematical notation to represent or illustrate these
biological ideas (and in mathematically modeling the systems these ideas are
representing or illustrating) is that mathematics takes on "a life of its
own". This is why my father often mused that mathematical languages are
complex!  But what can, and often does, happen is that the semantics (the
meaning) of the illustrations can be forgotten and the math will continue to
actively iterate on, going its merry way, but meaning nothing to the
original purpose anymore. I suspect that this may be exactly what has
happened with "Set Theory" and "Category Theory", as they are being defined
in this discussion. It's almost inevitable with notation modes that are as
useful and multi-applicable as these.

Judith

----- Original Message -----
From: "Tim Gwinn" <***>
To: <***>
Sent: Tuesday, June 15, 2004 10:35 PM
Subject: Re: [ROSEN] What is /are The Logic(s) of Life?... and Adjointness
is Fundamental in Categories and Topoi of Biological Systems


> > -----Original Message-----
> > From: ROSEN Forum [mailto:*** Behalf Of Ionel
> > Sent: Tuesday, June 15, 2004 9:27 PM
> > To: ***
> > Subject: Re: What is /are The Logic(s) of Life?... and Adjointness is
> > Fundamental in Categories and Topoi of Biological Systems
> >
> >
> > Hi, Tim:
> >
> > I am answering your specific questions one-by-one as we go along with
the
> > quotes.
> >
> > On Tue, 15 Jun 2004 13:19:40 -0400, Tim Gwinn <***> wrote:
>
> ---snip--
>
>
> > >> 3. Robert Rosen explicitely gives up Sets as UNSUITABLE for modeling
> > >> Complex Systems in his "Essays..." book published in 2001, and is in
> > favor of 'structured' Categories that aren't Sets.
> >
> > >
> > >TG: Can you give me the page reference for that? I can't seem to recall
> > that.>>
> > ------------------------------------------------------------
> >
> > ICB's Response: I will provide you with a precise page number,
> > but it would
> > be far better... that you find it... because you'd have then also the
> > benefit of Robert's original thinking behind this important choice.
> > -----------------------------------------------------------------
> --snip--
>
>
> Ionel,
>
> I have read Essays several times. I cannot recall seeing either where 1)
> Rosen rejects sets for relational modeling, or 2) mentions "'structured'
> Categories" (for relational modeling, or anything else). In chapter 17
"What
> Does It Take to Make an Organism?", where he discusses the (M,R)-system
> model in detail, he refers to the components in the model as "sets" and as
> "sets of mappings" (p. 261,262).
>
> Regards,
> Tim
>