Thanks for your article. It is, indeed, very clear and concise, even
without the diagrams. It some ways it is better than my review, in that it
is more concrete and less theoretical. But unfortunately I don't think it
addresses my main question.
You say:
> Rosen goes through an argument (that helps to have
>the accompanying relational diagrams) where it is shown that
>a finite mechanistic description is not possible with a truly
>complex system. An infinite regress forms if one uses only
>isolated separable parts. This is a definition of a complex
>system. It cannot be represented by a finite direct sum
>(synthetic) model where all the components can be divided
>into separate disjoint sets.
and this is all clear to me in the book. What is not clear to me, as I
discuss below, is how Rosen avoids the regress himself, even in
non-mechanistic systems. Now I know that in LI, Rosen does not provide a
full explanation of his closure, but only refers to "earlier publications"
in which he does (see p. 251). I presume that that is "Anticipatory
Systems". I REALLY regret that I have not read AS. It's been on my list for
years. At the moment, it's just too damned big and expensive, and I don't
have time for this any more. If one of you could sketch out the details,
I'd be much obliged.
So I cannot claim that he does not avoid the regress. I only do not
understand how he claims to. In my own interpretation (below), it may be
that I am myself committing a mechanistic reduction of his organismal
scheme. I understand that Don claims that final cause is sufficient to
overcome this problem. And I believe I understand the crucial diagram
itself, and in terms of causal categories. But what I do not understand is
how or if FORMALLY the regress is avoided, and in particular how or if
CATEGORY THEORY is instrumental in doing so.
Don:
===
Sorry, I was wrong to engage you at any emotional level. It distorts your
ability to reason. Under any ordinary circumstances, I would probably cut
off this argument. But, since, as you are apparently COMPLETELY incapable
of appreciating, I find Rosen and his perspective and argument FASCINATING
(in fact, obviously far too much, for all the time I've spent on this),
therefore I will continue. I really do have some specific questions, and I
have some hope that you or someone here can answer them, although there's
been little evidence of that so far. I will make every effort from now on
to simply ignore your insults.
Now so far your only answer to my question has been:
>In a complex system, final cause is invioked and lo and
>behold, the recursion is broken, due to the peculiar nature of
>final causality . . . The issue is not whether accepting
>final cause stops the recursion, it CLEARLY DOES, with or without
>category theory.
No formal argument, indeed, no argument, no explanationm, at all, merely an
assertion that the magic of final cause cures all.
So, I delve back into LI. The first key section is 5J on p. 134, "Augmented
Abstract Block Diagrams". Here he introduces the mapping-space notation and
examines the threat of regress. The next section 5K "Finality in Augmented
Abstract Block Diagrams" introduces final cause as an explanatory principle
in such diagrams, but does so still leaving the regression intact.
We must then turn to the capstone of the book, the punchline, section 10C,
"Relational Models of Organisms". On p. 251 he closes the relational
diagrams to efficient causation. As I remarked above, at this point I need
to read AS to complete his train of though.
So, I admit that I AM NOT CARRYING OUT ROSEN'S FORMAL ARGUMENT THE WAY HE
WOULD HAVE ME. All I have the capacity to do is cast Rosen's scheme on p.
251 in terms of recursive functional equations. When I do so, it APPEARS to
me that the regress persists. If I am wrong, please explain how.
To demonstrate this, I'm including here a short part of a letter I
exchanged with Turchin on Rosen. The full letter addresses analytic and
synthetic models and the final organism diagram, and I'm sending it by post
since the mathematics doesn't do well in ASCII here. Or, please find it in
postscript (some of the arrows might not come out perfectly) at
ftp://kong.gsfc.nasa.gov/pub/joslyn/92augtur.ps
and in self-contained LaTeX at
ftp://kong.gsfc.nasa.gov/pub/joslyn/92augtur.tex
I am also making my errata on the book similarly available at
ftp://kong.gsfc.nasa.gov/pub/joslyn/ros_err.ps
ftp://kong.gsfc.nasa.gov/pub/joslyn/ros_err.tex
In this strict ASCII format I can only reproduce a small part of the math,
but I will do so for the most important section, about how Rosen tried to
avoid regress in section 10C through category theory.
=== begin quotation
I think I can lay out the problem in function notation like this:
(**) A = { a }
B = { b } b:F -> \Phi b(f) = \phi
F = { f } f:A -> B f(a) = b
\Phi = { \phi } \phi:B -> F \phi(b) = f
$A$ must be given. But the other definitions are circular, e.g.\ $B$
depends on $F$, and $F$ on $B$. In Rosen's notation, we have
H(X,Y) = { g:X -> Y }
B = H(F,\Phi), F = H(A,B), \Phi = H(B,F).
When we try to expand say $F$, we [obviously] regress:
F = H(A,B) = H(A,H(F,\Phi))
So it seems that these functions cannot be well-specified (in set
theory), but perhaps well-CONSTRUCTED (in category theory or your
cybernetic machine theory).
Like you, I can derive $f'$:
\begin{eqnarray}
f' & = & \phi(b) \nonumber \\
& = & ( b(f) )( b ) \nonumber \\
& = & ( (f(a))(f) )( f(a) ) \label{final}
\end{eqnarray}
In \equ{final}, $a$ is used only as an argument to $f$, but each of the
symbols $f$ and $f(a)$ are used as {\em both} functions and arguments to
functions. This is what is captured explicitly by your $I_{AB}$ notation,
but is implicit here.
Am I correct in understanding you to mean that this is a standard
methodology in functional theory or recursive function theory? That is,
recursive spaces of functions as in \equ{recur} and functions as arguments
and arguments as functions are nothing special?
Can you hack out some quick LISP or REFAL code which implements this?
=== end quotation
It is the system (**) which Turchin found "uninteresting", although I am
not sure I do.
The question, again:
************************************************************************
Am I correct that system (**), WHEN LOOKED AT AS A SYSTEM OF FUNCTIONAL
EQUATIONS, still regresses?
Is there something about category theory in particular which avoid this?
************************************************************************
O----------------------------------------------------------------------------->
| Cliff Joslyn, NRC Research Associate, Cybernetician at Large, (301) 286-7816
| Code 522.3, NASA Goddard Space Flight Center, Greenbelt MD 20771 USA
| joslyn@kong.gsfc.nasa.gov http://groucho.gsfc.nasa.gov/joslyn/joslyn.html
V All the world is biscuit shaped. . .