Jeff:
====
>I may be wrong about this, but I think he is saying that in the world
>of biology, infinite regresses are here to stay. We shouldn't be
>trying to figure out how to get rid of them (by developing reductionist
>theory that doesn't look at the whole picture), we should be dealing
>with them...incorporating the infinite regresses into our theory.
Well, on the one hand, he notes the regress of state systems, and how
Newton's introduction of acceleration as a state variable truncates the
regress. He then goes on to note the potential regress in relational
systems, and how his introduction of the crucial level of closure of
efficient causation truncates the regress. In so doing, he claims to turn a
regress into a CYCLE of causal explanation (yes, Don, I understand
perfectly). Regresses and cycles are not the same, but they are related:
cycles can generate regresses when they are approached linearly and teased
apart; but regresses can result from other processes as well. That is what
I showed in the former post: unpacking Rosen's relational scheme for the
organism, which is itself cyclic, results in a regress. Newton's Taylor's
series expansion is also a regress, but does not result from a cycle.
Onar:
====
>If Rosen is right, why hasn't he come into the light of the scientific
>community?
As Don has said, Rosen is a highly respected and sought-after member of the
systems theoretical community, at least. I can't speak to "science" in
general. But, as I'm sure you appreciate, popularity during one's lifetime
is hardly the standard one would want to apply to assess scientific
validity or value.
Don:
====
I think that you think that I want to understand relational biology (RB)
from the reductionist mathematical view AS OPPOSED TO the holistic,
relational, gestalt view. In fact, Don, I WANT TO DO BOTH. The relational
whole is something I think I understand already. It's the math that I don't
get.
I understand and interpret Rosen at different levels and in different
ways. One can take Rosen from the point of view of philosophy, of
science, of systems theory, of category theory, of a god-damned
first-order cybernetician left-brained modern mathematical
reductionist (ahem :> ), indeed, of poetry.
Thinking it over, I believe that this is part of the reason why you liked
my review, but finds my later comments insufficient. In the review, I was
attempting to put the book forward in its own terms, and at many different
of these levels of language. Later on, I began to both develop a more
critical perspective, and also to work more at the level of the FORMAL
LANGUAGE.
My later questions were very specific, and directly related to the formal
level. Indeed, one of Rosen's strong claims (I can't find the page right
now) is that not only can we tease apart these elusive, complex relations
present in organisms, but we can do so in RB through FORMAL MODELING in a
way not done in other systems theories. In paritcular, he suggests that
category theory is the proper language, which has advantages over others
(e.g. recursive functions).
Let me cut to the mathematical chase and then get back to more general
issues:
>Notice also that your rendition of what is in the book reduces his
>content and meaning BY IGNORING THE FACT THAT HE NEEDS TWO KINDS OF
>ARROWS TO REPRESENT THE MAPPINGS. This should have been a clue.
OK, maybe. Perhaps I have a problem with extended block diagrams in
general. Let's consider the simple example on p. 250, with no closure or
recursion. My understanding is that the single arrow
f
A ---> B
means
f:A->B ($)
in function notation, where A and B are sets, and that that is equivalent
to the compound arrow:
f
/
/
@
A ---> B ($$)
where f--@A is the other color arrow. Then on p. 250 Rosen gives the
diagram
f
/ ^
/ \
@ \
A ---> B @--- Phi ($$$)
f Phi
which he claims is equivalent to A ---> B ---> H(A,B) where
H(A,B) = { A->B }
in function notation. But interpreting ($$$) in the function notation of
($) and ($$), I get f:A->B and Phi:B->f. But Phi:B->f cannot be correct,
since f is an ELEMENT of H(A,B), not a whole set as it must be to be in
the range of a mapping.
Now back to the general argument:
>Rosen accomplishes this
>by finding a particular f which has a purpose, which has an answer to
>the question why f? The trick is in that final cause means that SOME
>B gives the reason for some f! This allows amapping from a B to f
>and it closes the diagram.
>The entire
>argument above manifests itself in the turning around of one arrow in a
>diagram, resulting in its achieving closure and terminating the regression.
Yes, I think I understand all that at the level of English. What I find is
that real understanding doesn't come until you work through all the horrid
details of the math, though.
>Why this seems ironic with respect to our discussion is that our
>reductionist training blinds us to the significance of such a ploy.
Not mine. I, at least, feel capabale of talking and thinking about
something in more than one way. In this case, I've been stubborn to keep to
that way, and thought I made that clear (perhaps not). So when I ask "How
does category theory help RB?", don't think that I can't or don't or don't
want to understand RB in any other terms. On the contrary, I dwell on the
mathematical questions precicely BECAUSE I believe I understand RB in the
OTHER languages.
Nor is a syntactical, reductionistic approach a priori invalid. Don't
replace a dogma of modern reductionism with one of post-modern
contextualism. One powerful view, filtered through Rosen and Pattee, that I
have brought deeply into myself is that of COMPLEMENTARITY: there are many,
alternate, non-comparably valid ways of looking at anything complex. Your
"reductionist" insults are mere ad hominem.
>Rosen accomplishes this
>by finding a particular f which has a purpose, which has an answer to
>the question why f? The trick is in that final cause means that SOME
>B gives the reason for some f! This allows amapping from a B to f
>and it closes the diagram.
I presume that you understand that I understand all this already. But
again, this is in ENGLISH. Rosen claims a valid MATHEMATICAL model. This is
what I'm trying to understand.
>As I understand, you are looking for the above explanation to come from
>category theory or some other part of the logic in the argument.
Yes.
>This would
>be exactly what we'd expect in a syntactical, reductionist scheme.
Well, that may be. But you say:
>Once again, the answer is not in the formalism.
But that's exactly what Rosen PROMISES in his own theory.
>HOWEVER,
>Rosen brought this notion of anticipation and final cause in INDEPENDENT
>of the rest of the argument and merely uses category theory as a
>rigorous way of representing the interplay of the ideas.
It may be that category theory is not necessary for RB. Indeed, I think I
understand it without category theory. It is not clear to me that Rosen is
making this claim. But Rosen is surely claiming that you CAN express RB in
category theory, as distinct from recursive equations. And this is the
thrust of my questions.
> Now, forgive me, but I will avoid trying to unscramble the notation
>in the rest of the message until you tell me whether or not this
>answers your question.
No. I already understood everything above this. It's the math I don't get.
>Just to be
>argumentative, did you ever ponder the possibility that emotions
>compliment reason at times?
Of course. Depends on what you're doing. And how far out of balance you
get.
>(Don't you pity my students?)
Nah, you'd be a great teacher. Undergrad, at least. I don't think I'd want
you as a thesis advisor.
>I never learned a better way to operate, so maybe I can be
>eligible for some handicapped status?
OK, I'll check the ADA on your behalf.
>As I said above final cause does not always end a regression, only in
>complex systems.
Yes, the claim is for systems closed to efficient causation.
>We did find that one diagram was
>misprinted and the arrow of crucial interest was still poining in the
>old direction allthough the argument in the text said it was turned
>around and this of course had to be the one that ended the regression.
Yes, the crucial diagram is hopelessly full of typos. But I got by that.
>I say this because, as I recall, this was not in your list of errors.
I believe it was.
>> (**) 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
>
>O. K. I think I see what is troubling you. You are calling the very closure
>in the relational diagram which Rosen claims breaks the sequence of
>regression a regression!!!! In your description of the math, you correctly
>show that the diagram contains a cycle and therefore the same sequence of
>mappings occurs again and again. YES!!!! THAT'S THE POINT!!!
>In section you quote (and elsewhere, he shows that the diagrams MUST BE
>EXTENDED WITHOUT CLOSURE UNLESS THE CLOSING CYCLE CAN BE FORMED.
>The very piece of math you are calling recursive is evidence that the closing
>cycle has been formed and that the mapping goes around and around as it should.
>Is that now clear?
Yes, I think we understand each other at one level. But I think we disagree
about the significance. As I said above, it's obvious when you try to
linearlize a cycle you get a regress. That might be OK sometimes, but
instead it's better to look at the cyclic whole. But, you see, that
approach leads DIRECTLY to the recursive function theory you so disparage.
THIS is Turchin's view of the crucial (**), as a whole, cyclic set of
recursive functions, and thus, to him, uninteresting.
My real question is whether anything else is gained by using category
theory on (**).
>What you need to do
>is stop
>focusing on the mathematical expression for a cycle in a map and step
>back and see that the Newtonian ploy has been accomplished in relational
>(augmented block diagrams) notation.
Done that already. Is it OK if I try to do both, Don?
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| Cliff Joslyn, NRC Research Associate, Cybernetician at Large, (301) 286-7816
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