Category theory

[glen e. p. ropella]

So, I'm sitting here reading "Life Itself" by Rosen, and he keeps
going on and on about how complex systems can be represented by what
he calls "components" (that are defined via the variance of the
behavior in a system when that component is removed) and mappings
between the elements in one component and the elements in another.
I.e. let components A, B, and C, make up system S.  Define mappings f
and g as: f:A->B and g:B->C, where the mappings represent causal
entailment from one set to the other.  (This means that "if b = f(a),
where b in B and a in A, then a provides the material cause of f(a)
and f provides efficient cause of f(a).")

Now, he claims that one can define arbitrary "abstract block diagrams"
by assembling these sets via mappings between them.  For example, let
A, B, C, and D be components.  Let p, q, r, and s be mappings such
that:

    p: A->B
    q: AxB->C
    r: B->CxD
    s: D->A

This would create an "abstract block diagram" that looked like:

         +-----p-----+     +----r-----+
         |           |     |          |
         |           |     |     +----+----+
         |           |     |     |         |
         |          \|/    |    \|/       \|/
       +---+       +---+   |   +---+     +---+
       | A |---+---| B |---+   | C |     | D |
       +---+   |   +---+       +---+     +---+
        /|\    |                /|\        |
         |     +----------q------+         |
         |                                 |
         |                                 |
         +----------------s----------------+

(Mind you, this type of example is not in the book, at least not up to
where I've read.  Also note that Rosen's couching of this system would
be that the blocks A, B, C, and D are actually the mappings between
inputs s, p, q, and r and outputs p, q, r, and s, respectively.  But,
it's kinda like the necker cube in that the structure doesn't change.)

Now all this is fine and dandy.  But, what worries me is whether or
not this is subject to analysis.  E.g. in his simple example, where
f:A->B and "a is material cause of f(a)", I have no problems.  But, in
a diagram like the above, what might be the material cause of, say,
q(a,b)?  Must we say, "a *and* b provide material cause?"  Or can we
say things like "a is material cause of q(a,b)" and "b is material
cause of q(a,b)?"

Furthermore, it's not even clear what he means by "cartesian product"
in this sense.  In a normal sense, the cartesian product of two sets
means a matching of elements from one set with the elements of
another. E.g. 3d space is simply independently matched real numbers.
Something like |Cx|R gives rise to vectors like (a + bi, x), a,b,x in
|R.  In all cases the matching is unconstrained.  And that's what
makes |R3 a cartesian product.

This independence has not been established.  E.g. let A, B and C be
componenets and let f, g be mappings f:AxB->C and g:A->B, then for a
in A, b in B, and c in C, if b = g(a) and c = f(a,b), "a is material
cause for b and g is efficient cause for b" *and* "a and b are
material cause for c and f is efficient cause for c."  This places a
dependence of b upon a and g such that it's ambiguous whether b is
really needed at all, and we could get away with defining a new
mapping, h:A->C where c=h(a).

Furthermore, it hasn't even been established that sequentiality is not
necessary.  E.g. if A, B are components and f:A->A and g:A->B are
mappings and a1, a2 in A, then let a2 = f(a1) and b = g(f(a1)) =
g(a2), but g(a2) != g(a1). Then it's not at all clear if "a1 is
material cause of b" unless it f is applied before g.

I haven't done any exploring into category theory; but, it's my
intuition that that won't be necessary to show these flaws in Rosen's
ideas. (I'm presuming that they're flaws... if I give him the benefit
of the doubt at such an early stage, it's likely that his flaw will
become my flaw later on. [grin])

Any thoughts?

Thanks.
glen
p.s. For those of you who are wondering where in hell this came
from, check:
http://www.santafe.edu/projects/swarm/archive.modelling/list-archive.9705/0046.html
-- 
{glen e. p. ropella <address@hidden> |  Send lawyers, guns, and money!  }
{Hive Drone, SFI Swarm Project         |            Hail Eris!            }
{http://www.trail.com/~gepr/home.html  |               =><=               }


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